Nicolas De Saxcé

LINEAR RANDOM WALKS ON THE TORUS

Keywords: 2010 Mathematics Subject Classification. Primary Sum-product, Random walk, Toral automorphism

We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption. An application is given to expansion in simple groups, modulo arbitrary integers. BFLM [9, Theorem A]. thm:main Theorem 1.2 (Quantitative equidistribution on the torus). Under the assumptions of Theorem thm:easy 1.1, let λ 1 denote the top Lyapunov exponent associated to µ.

Introduction

The goal of the present paper is to study the equidistribution of linear random walks on the torus. We are given a probability measure µ on the group SL d (Z) of integer matrices with determinant one, and consider the associated random walk (x n ) n≥0 on the torus T d = R d /Z d , starting from a point x 0 in T d , and moving at step n following a random element g n with law µ:

x n = g n x n-1 = g n . . . g 1 x 0 .

We say that the measure µ on SL d (Z) has some finite exponential moment if there exists ε > 0 such that

g ε dµ(g) < ∞,
where denotes an arbitrary norm on M d (R), the space of d × d matrices with real coefficients. Our goal is to prove the following theorem. Then, for every irrational point x in T d , the sequence of measures (µ * n * δ x ) n≥1 converges to the Haar measure in the weak- * topology.

With an additional proximality assumption, this theorem was proved a decade ago by Bourgain, Furman, Lindenstrauss and Mozes BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], and we follow their approach to this problem, via a study of the Fourier coefficients of the law at time n of the random walk on T d . One advantage of this method -besides being the only one available at the present -is that it yields a quantitative statement, giving a speed of convergence of the random walk, in terms of the diophantine properties of the starting point x, see Given λ ∈ (0, λ 1 ), there exists a constant C = C(µ, λ) > 0 such that for every x ∈ T d and every t ∈ (0, 1/2), if for some a ∈ Z d \ {0}, | µ * n * δ x (a)| ≥ t and n ≥ C log a t , then there exists q ∈ Z >0 and x ∈ ( 1 q Z d )/Z d such that q ≤ Ct -C and d(x, x ) ≤ e -λn .

One of our motivations for removing any proximality assumption from this theorem was to generalize a theorem of Bourgain and Varjú on expansion in SL d (Z/qZ), where q is an arbitrary integer, to more general simple Q-groups. We briefly describe this application at the end of the paper, in § ss:exp 6.1.

In BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], the proximality assumption is used at several important places, especially in the study of the large scale structure of Fourier coefficients BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]Phase I]. Let us mention the main ingredients we had to bring into our proof in order to overcome this issue.

One important tool in the proof of Bourgain, Furman, Lindenstrauss and Mozes BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] is a discretized projection theorem, due to Bourgain Bourgain2010 [8, Theorem 5], giving information on the size of the projections of a set A ⊂ R d to lines. But, when the random walk is not proximal, one should no longer project the set to lines, but to subspaces whose dimension equals the proximality dimension of Γ. One approach, of course, would be to generalize Bourgain's theorem to higher dimensions, and this has been worked out by the first author he_projection [?]. But it turns out that the natural generalization of Bourgain's theorem, used with the general strategy of BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], only allows to deal with some special cases he_schubert [?]. Here we take a different route. Instead of working in the space Z d of Fourier coefficients, we place ourselves in the simple algebra E ⊂ M d (R) generated by the random walk. This allows us to use the results of the first author on the discretized sum-product phenomenon in simple algebras He2016 [START_REF] He | Discretized sum-product estimates in matrix algebras[END_REF]. Thus, instead of a projection theorem, we use a result on the Fourier decay of multiplicative convolutions in simple algebras, derived in Section sc:sumprod 2, and generalizing a theorem of Bourgain for the field of real numbers Bourgain2010 [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF]Theorem 6]. Then, in order to be able to apply this Fourier estimate to the law at time n of the random walk, we have to check some non-concentration conditions. For that, we use a result of Salehi Golsefidy and Varjú SGV [START_REF] Golsefidy | Expansion in perfect groups[END_REF] on expansion modulo prime numbers in semisimple groups, combined with a rescaling argument, proved with the theory of random walks on reductive groups. In the end, we obtain some Fourier decay theorem for the law at time n of a random walk on SL d (Z), Theorem thm:decay 3.19, which, we believe, bears its own interest and, we hope, will have other applications.

The rest of the proof, corresponding to BFLM [9, Phase II], follows more closely the strategy of BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]. But since at several points we had to find an alternative proof to avoid the use of the proximality assumption, we chose to include the whole argument, rather than refer the reader to BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]. We hope that this will make the proof easier to follow.

2. Sum-product, L 2 -flattening and Fourier decay sc:sumprod

The main objective of this section is to prove that in a simple real algebra, multiplicative convolutions of non-concentrated measures admit a polynomial Fourier decay. The precise statement is given in Theorem From now on, E will denote a finite-dimensional real associative simple algebra, endowed with a norm . Given a finite Borel measure µ on E and an integer s ≥ 1, we write

µ * s = µ * • • • * µ s times
for the s-fold multiplicative convolution of µ with itself. In order to ensure the Fourier decay of some multiplicative convolution of the measure µ, we need two assumptions: First, µ should not be concentrated around a linear subspace of E, and second, µ should not give mass to elements of E that are too singular.

To make these requirements more precise, let us set up some notation. For ρ > 0 and x ∈ E, let B E (x, ρ) denote the closed ball in E of radius ρ and centered at x. For a subset W ⊂ E, let W (ρ) denote the ρ-neighborhood of W ,

W (ρ) = W + B E (0, ρ).
For a ∈ E define det E (a) to be the determinant of the endomorphism E → E, x → ax. Note that since E is simple this quantity is equal to the determinant of E → E, x → xa. For ρ > 0, define S E (ρ), the set of badly invertible elements of E, as

S E (ρ) = {x ∈ E | |det E (x)| ≤ ρ}.
thm:fourier Theorem 2.1 (Fourier decay of multiplicative convolutions). Let E be a normed simple algebra over R of finite dimension. Given κ > 0, there exists s = s(E, κ) ∈ N and ε = ε(E, κ) > 0 such that for any parameter τ ∈ (0, εκ) the following holds for any scale δ > 0 sufficiently small. Let µ be a Borel probability measure on E.

Assume that (i) µ E \ B E (0, δ -ε ) ≤ δ τ ; (ii) for every x ∈ E, µ(x + S E (δ ε )) ≤ δ τ ; (iii) for every ρ ≥ δ and every proper affine subspace W ⊂ E, µ(W (ρ) ) ≤ δ -ε ρ κ . Then for all ξ ∈ E * with ξ = δ -1 , | µ * s (ξ)| ≤ δ ετ .

For E = R, this is due to Bourgain Bourgain2010 [8, Lemma 8.43]. Li proved in Li2018 [START_REF] Li | Discretized Sum-product and Fourier decay in R n[END_REF] a similar statement for the semisimple algebra R ⊕ • • • ⊕ R. While a more general statement should hold for any semisimple algebra, we do not pursue in this direction and focus in the present paper only on simple algebras.

2.1. L 2 -flattening. The aim of this subsection is to prove a sum-product L 2flattening lemma for simple algebras.

We shall consider both additive and multiplicative convolutions between measures or functions on E. To avoid confusion, we shall use the usual symbol * to denote multiplicative convolution and the symbol to denote additive convolution. In the same fashion, for finite Borel measures µ and ν on E, we define µ ν to be the push forward measure of µ ⊗ ν by the map (x, y) → x -y.

For a Borel set A ⊂ E, denote by |A| the Lebesgue measure of A. For δ > 0, define P δ = |B E (0, δ)| -1 1 B E (0,δ) . For absolutely continuous measures such as µ P δ , by abuse of notation, we write µ P δ to denote both the measure and the Radon-Nikodym derivative. For x ∈ E, we write D x to denote the Dirac measure at the point x. For K > 1, define the set of well invertible elements of E as

G E (K) = {x ∈ E × | x , x -1 ≤ K}.
Note that if x ∈ G E (K), the left, or right, multiplication by x as a map from E to itself is O(K)-bi-Lipschitz.

pr:apla Proposition 2.2 (L 2 -flattening). Let E be a normed finite-dimensional simple algebra over R of dimension d ≥ 2. Given κ > 0, there exists ε = ε(E, κ) > 0 such that the following holds for δ > 0 sufficiently small. Let µ be a Borel probability measure on E. Assume that (i) µ is supported on G E (δ -ε );

(ii) δ -κ ≤ µ P δ 2 2 ≤ δ -d+κ ;

it:flatnc2

(iii) for every proper linear subspace W < E, ∀ρ ≥ δ, µ(W (ρ) ) ≤ δ -ε ρ κ .

Then, (µ * µ µ * µ) P δ 2 ≤ δ ε µ P δ 2 .

Remark 1. If E = R the same holds if condition it:flatnc2

(iii) is replaced by eq:flatnc1 eq:flatnc1

(2.1) ∀ρ ≥ δ, ∀x ∈ E, µ(B E (x, ρ)) ≤ δ -ε ρ κ .

Note also that when dim(E) ≥ 2, property ( eq:flatnc1

2.1) is implied by condition it:flatnc2

(iii).

rk:mu1half

Remark 2. We shall apply this proposition to measures that are not probability measures. It is clear that by making ε slightly smaller, the same statement holds for measures µ with total mass µ(E) ≥ 1 2 or just µ(E) ≥ δ ε . Proof. In this proof, the implied constants in the Landau or Vinogradov notation depend on the algebra structure of E as well as the choice of norm on it. We use the following rough comparison notation : for positive quantities f and g, we write

f g if f ≤ δ -O(ε) g and f ∼ g for f g and g f . For instance, if a ∈ G E (δ -ε ) then |det E (a)| ∼ 1.
To simplify notation, we shall also use the shorthand µ δ = µ P δ . Now assume for a contradiction that the conclusion of the proposition does not hold, namely eq:counterFlat0 eq:counterFlat0

(2.2) (µ * µ µ * µ) P δ 2 ≥ δ ε µ δ 2 .
Step 0: Compare the L 2 -norms of (µ * µ µ * µ) P δ and µ * µ δ µ δ * µ. For x ∈ E, write

(µ * µ µ * µ P δ )(x) =|B(0, δ)| -1 µ ⊗4 {(a, b, c, d) | ab -cd ∈ B E (x, δ)} ≤|B(0, δ)| -1 (µ ⊗4 ⊗ P ⊗2 δ ){(a, b, c, d, y, z) | a(b + y) -(c + z)d ∈ B E (x, δ 1-2ε )} |B(0, δ 1-2ε )| -1 (µ ⊗4 ⊗ P ⊗2 δ ){(a, b, c, d, y, z) | a(b + y) -(c + z)d ∈ B E (x, δ 1-2ε )} =(µ * µ δ µ δ * µ P δ 1-2ε )(x).
Above at the sign ≤, we used the assumption that Supp(µ) ⊂ B E (0, δ -ε ). Therefore, by Young's inequality,

µ * µ µ * µ P δ 2 µ * µ δ µ δ * µ P δ 1-2ε 2 ≤ µ * µ δ µ δ * µ 2 .
To conclude step 0, we deduce from the above and ( eq:counterFlat0

2.2) that

eq:counterFlat eq:counterFlat

(2.3) µ * µ δ µ δ * µ 2 µ δ 2 .
Step 1: Discretize the measure µ using dyadic level sets. For a subset A ⊂ E, we denote by N (A, δ) the least number of balls of radius δ in E that is needed to cover A. By a δ-discretized set we mean a union of balls of radius δ.

Note that if A is a δ-discretized set then N (A, δ) E δ -d |A|.
Is is easy to check that there exist δ-discretized sets A i ⊂ B E (δ -ε ), i ≥ 0 such that A i is empty for i log 1 δ , and eq:mullAi eq:mullAi

(2.4) µ δ i≥0 2 i 1 Ai µ 3δ + 1.
Step 2: Pick a popular level in order to transform ( eq:counterFlat 2.3) into a lower bound on the additive energy between two δ-discretized sets. We have

µ * µ δ µ δ * µ = E×E (D a * µ δ ) (µ δ * D b ) dµ(a) dµ(b).
From the left inequality in ( eq:mullAi

2.4), µ * µ δ µ δ * µ i,j≥0 2 i+j (D a * 1 Ai ) (1 Aj * D b ) dµ(a) dµ(b). Observe that D a * 1 Ai = |det E (a)| -1 1 aAi and 1 Aj * D b = |det E (b)| -1 1 Aj b . Hence µ * µ δ µ δ * µ i,j≥0 2 i+j 1 aAi 1 Aj b |det E (a) det E (b)| dµ(a) dµ(b).

By (

eq:counterFlat 2.3), the triangular inequality and the assumption that µ is supported on

G E (δ -ε ), i,j≥0 2 i+j 1 aAi 1 Aj b 2 dµ(a) dµ(b) µ δ 2 ,
There are at most O(log 1 δ ) 2 1 terms in this sum. Hence by the pigeonhole principle, there exist i ≥ 0 and j ≥ 0 such that eq:defiandj eq:defiandj

(2.5) 2 i+j 1 aAi 1 Aj b 2 dµ(a) dµ(b) µ δ 2 .
From now on we fix such i and j. By the right-hand inequality in ( eq:mullAi 2.4), we find

2 i |A i | 1/2 = 2 i 1 Ai 2 µ 3δ 2 + 1 µ δ 2 ,
so that for all a, b ∈ G E (δ -O(ε) ), eq:aAi2leqmu eq:aAi2leqmu

(2.6)

2 i 1 aAi 2 µ δ 2 and 2 j 1 Aj b 1 1.
Hence by Young's inequality,

∀a, b ∈ G E (δ -O(ε) ), 2 i+j 1 aAi 1 Aj b 2 µ δ 2 .
This combined with ( eq:defiandj 2.5) implies that the set

B 0 = {(a, b) ∈ G E (δ -O(ε) ) ×2 | 2 i+j 1 aAi 1 Aj b 2 ≥ δ O(ε) µ δ 2 }
has measure µ ⊗ µ(B 0 ) 1. For c = (a, b) ∈ B 0 , using ( eq:aAi2leqmu

2.6), we find

1 aAi 1 Aj b 2 |aA i | 1/2 |A j b|,
and switching the role of aA i and A j b,

1 aAi 1 Aj b 2 |aA i ||A j b| 1/2 .
Hence,

1 aAi 1 Aj b 2 2 |aA i | 3/2 |A j b| 3/2 .
Note that aA i and A j b are δ 1-O(ε) -discretized sets. Hence the last inequality translates to eq:energyAiAj eq:energyAiAj

(2.7) E δ (aA i , -A j b) N (aA i , δ) 3/2 N (A j b, δ) 3/2
where E δ denotes the additive energy at scale δ, as defined in We are going to use Rusza calculus. For subsets A, A ⊂ E we write A ≈ A if

N (A -A , δ) N (A, δ) 1/2 N (A , δ) 1/2 .

Ruzsa's triangular inequality and the Plünnecke-Ruzsa inequality

TaoVu [START_REF] Tao | Additive combinatorics[END_REF]Chapters 2 & 6] can be summarized as : the relation ≈ is transitive 1 , i.e. A ≈ A and A ≈ A implies A ≈ A . By Tao's non-commutative version of the Balog-Szemerédi-Gowers lemma

Tao2008

[33, Theorem 6.10] applied to ( eq:energyAiAj 2.7), for every c ∈ B 0 , there exists A c ⊂ A i and A c ⊂ A j such that N (A c , δ) N (A i , δ) and N (A c , δ) N (A j , δ) and eq:aAcApcb eq:aAcApcb

(2.8) aA c ≈ A c b.
By taking δ-neighborhoods if necessary, we may assume that A c and

A c are δ-discretized sets. Write X = A i × A j ⊂ R 2d and X c = A c × A c ⊂ X. From the Cauchy-Schwarz inequality applied to the function x → B0 1 Xc (x) dµ ⊗2 (c), we infer that B0×B0 |X c ∩ X d | dµ ⊗2 (c) dµ ⊗2 (d) |X|.
By the pigeonhole principle, there exists c ∈ B 0 and

B 1 ⊂ B 0 such that µ ⊗2 (B 1 ) µ ⊗2 (B 0 ) 1 and for all c ∈ B 1 , |X c ∩ X c | |X|.
Abbreviate A c as A and A c as A . We then have, for every c ∈ B 1 , eq:AstarCap eq:AstarCap (2.9)

N (A ∩ A c , δ) N (A i , δ) and N (A ∩ A c , δ) N (A j , δ).
For c = (a, b) ∈ B 1 , by the Rusza calculus and ( eq:aAcApcb 2.8),

aA c ≈ A c b ≈ aA c . Since a ∈ G E (δ -ε ), this implies A c ≈ A c .

Using (

eq:AstarCap 2.9) and the definition of the symbol ≈, we get A ∩ A c ≈ A c and for the same reason

A ∩ A c ≈ A . Hence aA ≈ a(A ∩ A c ) ≈ aA c ≈ A c b ≈ (A ∩ A c )b ≈ A b.
(2.10) N (a -1 aA -A b -1 b, δ) N (A , δ), ∀(a, b) ∈ B 1 ∪ {(a , b )}.
Step 5: Apply the sum-product theorem stated below as Proposition pr:sumprod 2.3. We claim that the assumptions of Proposition pr:sumprod 2.3 are satisfied by the set A , the set B 1 and the measure µ for the parameters κ/2 in the place of κ and O(ε) in the place of ε. Indeed, using Young's inequality and remembering ( eq:aAi2leqmu 2.6), we obtain

µ δ 2 2 i+j 1 a Ai 1 Aj b ≤ 2 i |a A i | 1/2 2 j |A j b | µ δ 2 . Hence 2 i |A i | 1/2 ∼ µ δ 2 and 2 j |A j | ∼ 1.
Inversing the roles of A i and A j , we get also 2 i |A i | ∼ 1. Thus, eq:2iAimu2 eq:2iAimu2

(2.11) |A i | ∼ µ δ -2 2 and 2 i ∼ µ δ 2 2 . Hence |A | |A i | µ δ -2 2 ≤ δ κ , which implies, as A is δ-discretized, N (A , δ) δ -d+κ .
Moreover, let ρ ≥ δ and x ∈ E and let B = B(x, ρ). Since µ 3δ = µ P 3δ , inequality ( eq:flatnc1 2.1) implies µ 3δ (B) ρ κ .

1 Strictly speaking, ≈ is not relation, because it involves an implicit constant in the notation.

By (

eq:2iAimu2

2.11) and (

eq:mullAi 2.4),

|A i ∩ B| |A i | 2 i |A i ∩ B| µ 3δ (B) . But A ⊂ A i and |A | |A i |, hence |A ∩ B| |A | |A i ∩ B| |A i | ρ κ .
It follows that for all ρ ≥ δ,

N (A , ρ) ρ -κ .

The verification of the other assumptions in Proposition

pr:sumprod 2.3 are straightforward, so we can apply Proposition pr:sumprod 2.3, which leads to a contradiction to ( eq:countersumprod 2.10) when ε > 0 is chosen small enough.

2.2.

A sum-product theorem. In the proof of L 2 -flattening, we used the following result.

pr:sumprod Proposition 2.3 (Sum-product estimate in simple algebras). Let E be a normed finite-dimensional simple algebra over R of dimension d ≥ 2. Given κ > 0, there exists ε = ε(E, κ) > 0 such that the following holds for every δ > 0 sufficiently small. Let A be a subset of E, µ a probability measure on E, and B a subset of

E × E. Assume (i) A ⊂ B E (0, δ -ε ); (ii) ∀ρ ≥ δ, N (A, ρ) ≥ δ ε ρ -κ ; (iii) N (A, δ) ≤ δ -(d-κ) ; it:spmu1 (iv) µ is supported on G E (δ -ε ); it:spmu3 (v) for every proper linear subspace W < E, ∀ρ ≥ δ, µ(W (ρ) ) ≤ δ -ε ρ κ ; it:spB (vi) µ ⊗ µ(B) ≥ δ ε . Then for every a , b ∈ G E (δ -ε ), there exists (a, b) ∈ B ∪ {(a , b )} such that N (a -1 aA + Ab -1 b, δ) ≥ δ -ε N (A, δ).
The idea of the proof is to consider the action of E × E on E by left and right multiplication and to apply a sum-product theorem He2016 [START_REF] He | Discretized sum-product estimates in matrix algebras[END_REF]Theorem 3] for irreducible linear actions due to the first author. For the reader's convenience, let us recall the statement of the latter. thm:sumprod_he Theorem 2.4 (Sum-product theorem for irreducible linear actions). Given a positive integer d and a real number κ > 0 there exists ε = ε(d, κ) > 0 such that the following holds for δ > 0 sufficiently small. Let X be a subset of the Euclidean space R d and Φ ⊂ End(R d ) a subset of linear endomorphisms. Assume

it:spX1 (i) X ⊂ B R d (0, δ -ε ); (ii) for all ρ ≥ δ, N (X, ρ) ≥ δ ε ρ -κ ; (iii) N (X, δ) ≤ δ -(d-κ) ; it:spPhi1 (iv) Φ ⊂ B End(R d ) (0, δ -ε ); it:spPhi2 (v) for all ρ ≥ δ, N (Φ, ρ) ≥ δ ε ρ -κ ; it:spPhi3 (vi) for every proper linear subspace W ⊂ R d , there is ϕ ∈ Φ and w ∈ B W (0, 1) such that d(ϕw, W ) ≥ δ ε . Then N (X + X, δ) + max ϕ∈Φ N (X + ϕX, δ) ≥ δ -ε N (X, δ).
Here, of course, ϕX denote the set {ϕx | x ∈ X}

Proof of Proposition

pr:sumprod 2.3. In this proof the implied constants in the Vinogradov or Landau notation may depend on E. We may assume without loss of generality that B ⊂ Supp(µ) × Supp(µ). This implies that for all (a, b) ∈ B, a , a -1 , b , b -1 ≤ δ -ε , and consequently the multiplication on the left or right by a or b is a δ

-O(ε) -bi- Lipschitz endomorphism of E. For (a, b) in B, define ϕ(a, b) ∈ End(E) by ∀x ∈ E, ϕ(a, b)x = -a -1 a xb -1 b.
We would like to apply the previous theorem to

X = A and Φ = {ϕ(a, b) ∈ End(E) | a, b ∈ B}.
We claim that the assumptions of Theorem thm:sumprod_he 2.4 hold with O(ε) in the place of ε. Hence there is ε 1 > 0 such that when ε > 0 is small enough, we have either N (A + A, δ) ≥ δ -ε1 N (A, δ) in which case we are done, or there exists (a, b) ∈ B such that

N (A + ϕ(a, b)A, δ) ≥ δ -ε1 N (A, δ).
In the latter case we conclude by multiplying the set above by a -1 a on the left, (vi) fails with δ Cε in the place of δ ε . Namely, there is a linear subspace

N (a -1 aA -Ab -1 b, δ) ≥ δ O(ε) N (A + ϕ(a, b)A, δ) ≥ δ -ε1+O(ε) N (A, δ).

It remains to check the assumptions in Theorem

0 ∈ E such that µ(B 1 (b 0 )) ≥ δ ε . From the inequalities a -a a a a -1 -a -1 and a -1 -a -1 = ϕ(a, b 0 )-ϕ(a , b 0 ) (a -1 b -1 0 b ) ≤ ϕ(a, b 0 )-ϕ(a , b 0 ) a -1 b -1 0 b , we see that the map a → ϕ(a, b 0 ) is δ -O(ε) -bi-Lipschitz on B 1 (b 0 ).
W 0 ⊂ R d of intermediate dimension 0 < k < d such that ∀(a, b) ∈ B, d(W 0 , ϕ(a, b)W 0 ) ≤ δ Cε , where d denotes the distance on the the Grassmannian Grass(k, d) of k-planes in R d defined by d(W, W ) = min w∈B W (0,1) d(w, W ) = min w ∈B W (0,1) d(w , W ).
In particular, for a, a ∈ B 1 (b 0 ), we have

d(W 0 , ϕ(a, b 0 )W 0 ) ≤ δ Cε and d(W 0 , ϕ(a , b 0 )W 0 ) ≤ δ Cε .
Multiplying the second inequality on the left by a -1 a , we obtain

d(a -1 a W 0 , ϕ(a, b)W 0 ) ≤ δ (C-O(1))ε .
By the triangular inequality,

d(W 0 , a -1 a W 0 ) ≤ δ (C-O(1))ε ,
which means eq:dgwW0 eq:dgwW0

(2.12)

∀g ∈ a -1 B 1 (b 0 ), ∀w ∈ W 0 , d(gw, W 0 ) ≤ gw d(gW 0 , W 0 ) ≤ δ (C-O(1))ε w .
Observe that the assumption it:spmu3

(v) of Proposition pr:sumprod 2.3 implies that the subset B 1 (b 0 ) ⊂ E is δ O(ε) -away from linear subspaces. Hence so is the subset a -1 B 1 (b 0 ). Using He2016 [START_REF] He | Discretized sum-product estimates in matrix algebras[END_REF]Lemma 8], we obtain from ( eq:dgwW0 2.12),

∀x ∈ B E (0, 1), ∀w ∈ B W0 (0, 1), d(xw, W 0 ) ≤ δ (C-O(1))ε .
We can do the same argument for the right multiplication. Thus, similarly,

∀x ∈ B E (0, 1), ∀w ∈ B W0 (0, 1), d(wx, W 0 ) ≤ δ (C-O(1))ε .
Consider the map f : Grass(k, d) → R defined by

f (W ) = B E (0,1)×B W (0,1) d(xw, W ) + d(wx, W ) dx dw.
On the one hand, from the above, f (W 0 ) ≤ δ (C-O(1))ε . On the other hand, f is continuous and defined on a compact set. It never vanishes for the reason that a zero of f must be a two-sided ideal of E contradicting the simplicity of E. Hence f has a positive minimum on Grass(k, d). We obtain a contradiction if C is chosen large enough, proving our claim regarding item it:spPhi3 (vi).

2.3.

Fourier decay for multiplicative convolutions. The goal here is to prove Theorem thm:fourier 2.1 using iteratively the L 2 -flattening lemma proved above. Let E be any finite-dimensional real algebra. The Fourier transform of a finite Borel measure µ on E is the function on the dual E * given by

∀ξ ∈ E * , μ(ξ) = E e(ξx) dµ(x).
where e(t) = e 2πit for t ∈ R, and we simply write E * × E → R; (ξ, x) → ξx for the duality pairing. The product on E yields a natural right action of E on E * given by ∀ξ ∈ E * , x ∈ E, y ∈ E, (ξy)(x) = ξ(yx), and for finite Borel measures µ and ν on E, the Fourier transform of their multiplicative convolution is given by eq:hatmuffnu eq:hatmuffnu

(2.13) µ * ν(ξ) = ν(ξy) dµ(y).
The idea of the proof of Theorem thm:fourier 2.1 is to iterate Proposition pr:apla 2.2 to get a measure with small L 2 -norm, and then to get the desired Fourier decay by convolving one more time. Two technical issues arise. First, after each iteration, the measure we obtain does not necessarily satisfy the non-concentration property required by Proposition pr:apla 2.2. To settle this, at each step, we truncate the measure to restrict the support on well-invertible elements. Second, the measure we obtain in the end of the iteration is not an additive convolution of a multiplicative convolution of µ but some measure obtained from µ through successive multiplicative and additive convolutions. To conclude we need to clarify relation between the Fourier transforms of these measures. This is settled in Lemma lm:ordre 2.7. lm:iterNC Lemma 2.5. Let E be a finite-dimensional normed algebra over R and µ a Borel probability measure on E such that for some τ, ε > 0,

(i) µ E \ B E (0, δ -ε ) ≤ δ τ ; (ii) for every x ∈ E, µ(x + S E (δ ε )) ≤ δ τ ;
(iii) for every ρ ≥ δ and every proper affine subspace W ⊂ E, µ(W (ρ) ) ≤ δ -ε ρ κ . Set µ 1 = µ |B E (0,δ -ε ) and define recursively for integer k ≥ 1,

η k = µ k|E\S E (δ 2 k ε ) and µ k+1 = η k * η k η k * η k .
Then we have for k ≥ 1, eq:iterNC0 eq:iterNC0

(2.14) µ k (E) ≥ 1 -O k (δ τ )
eq:iterNC1 eq:iterNC1

(2.15)

Supp(µ k ) ⊂ B E (0, δ -O k (ε) )
eq:iterNC2 eq:iterNC2

(2.16) ∀x ∈ E, µ k (x + S E (δ 2 k ε )) ≤ δ τ
eq:iterNC3 eq:iterNC3

(2.17)

∀ρ ≥ δ, ∀W ⊂ E proper affine subspace, µ k (W (ρ) ) ≤ δ -O k (ε) ρ κ .
As a consequence, the same holds for η k in the place of µ k .

Proof. The proof goes by induction on k.

The result is clear for k = 1, by assumption on µ. Assume ( eq:iterNC0

2.14)-(

eq:iterNC3

2.17) true for some k ≥ 1, so that the same holds for η k . Then ( eq:iterNC0 2.14) and ( eq:iterNC1 2.15) for k + 1 follow immediately.

Let us prove ( eq:iterNC2

2.16) for k + 1. Let x ∈ E. Since µ k+1 = η k * η k η k * η k , µ k+1 x+S E (δ 2 k+1 ε ) = η k {y ∈ E | |det E (yz-w-x)| ≤ δ 2 k+1 ε } dη k (z) d(η k * η k )(w) Note that for z ∈ Supp(η k ), by definition, |det E (z)| ≥ δ 2 k ε . Hence |det E (yz -w - x)| ≤ δ 2 k+1 ε implies y -(w + x)z -1 ∈ S E (δ 2 k ε ).
Therefore, by induction hypothesis ( eq:iterNC2

2.16)

µ k+1 x + S E (δ 2 k+1 ε ) ≤ max z∈Supp(η k ), w∈E η k (w + x)z -1 + S E (δ 2 k ε ) ≤ δ τ .
Finally, let us prove ( eq:iterNC3 2.17) for k + 1. Let ρ ≥ δ and let W be a proper affine subspace of E. We have as above

µ k+1 W (ρ) ≤ max z∈Supp(η k ), w∈E η k (w + W (ρ) )z -1 .
For all z ∈ Supp(η k ), we have |det ε) and by the induction hypothesis

E (z)| ≥ δ O k (
z ≤ δ -O k (ε) . Hence z -1 ≤ δ -O k (ε) . Thus (w + W (ρ) )z -1 = wz -1 + W z -1 + B E (0, ρ)z -1 ⊂ wz -1 + W z -1 + B E (0, δ -O k (ε) ρ),
which is nothing but the (δ -O k (ε) ρ)-neighborhood of another proper affine subspace.

Hence by induction hypothesis ( eq:iterNC3 2.17),

µ k+1 W (ρ) ≤ δ -O k (ε) ρ κ .
This finishes the proof of the induction step and that of the lemma.

lm:L2Fourier

Lemma 2.6. Let E be normed algebra over R of dimension d. Let µ and ν be Borel probability measures on E. Assume (i) µ P δ 2 2 ≤ δ κ , (ii) for every ρ ≥ δ and every proper affine subspace

W ⊂ E, ν(W (ρ) ) ≤ δ -ε ρ κ . Then for ξ ∈ E * with δ -1+ε ≤ ξ ≤ δ -1-ε , | µ * ν(ξ)| ≤ δ 2κ d+3 -O(ε)
Proof. This is a slightly more general form of Then for every integer m ≥ 1, for every ξ ∈ E * , the Fourier coefficient µ * m (ξ) is real and eq:ordre eq:ordre

(2.18) | ν * m (ξ)| (2 ) m ≤ µ * m (ξ).
The same inequality also holds for a finite Borel measure with total mass ν(E) ≤ 1.

Proof. We proceed by induction on m. For m = 1,

μ(ξ) = |ν(ξ)| 2 .
Assume then that ( eq:ordre 2.18) is true for some m ≥ 1. By ( eq:hatmuffnu 2.13), the Hölder inequality and the induction hypothesis,

| ν * (m+1) (ξ)| (2 ) m = ν * m (ξy) dν(y) (2 ) m ≤ | ν * m (ξy)| (2 ) m dν(y) ≤ µ * m (ξy) dν(y) = e(ξyx) dµ * m (x) dν(y)
Taking the 2 -th power and using again the Hölder inequality and ( eq:hatmuffnu 2.13), we obtain

| ν * (m+1) (ξ)| (2 ) m+1 ≤ e(ξyx) dν(y) dµ * m (x) 2 ≤ e(ξyx)) dν(y) 2 dµ * m (x) = e ξ(y 1 + • • • + y -y +1 -• • • -y 2 )x dν(y 1 ) . . . dν(y 2 ) dµ * m (x) = e(ξzx) dµ(z) dµ * m (x) = µ * (m+1) (ξ)
This proves the induction step and finishes the proof of ( eq:ordre 2.18). If ν is a finite Borel measure with ν(E) ≤ 1, we may apply ( eq:ordre

2.18) to the probability measure ν(E) -1 ν, which yields | ν * m (ξ)| ν(E) m (2 ) m ≤ µ * m (ξ) ν(E) 2 m .

Proof of Theorem

thm:fourier

2.1. Let d = dim(E).
Let ε 1 > 0 be the constant given by Proposition pr:apla 2.2 applied to the parameter κ/2. Define s max = d ε1 . First, remark that by the non-concentration assumption, µ P δ ∞ ≤ δ -d+κ-ε . Hence

µ P δ 2 2 ≤ µ P δ 1 µ P δ ∞ ≤ δ -d+κ/2
if we choose ε ≤ κ/2. For k = 1, . . . , s max , let µ k and η k be defined as in Lemma lm:iterNC 2.5. Since k ≤ s max is bounded, the implied constants in the O k (ε) notations in Lemma lm:iterNC 2.5 can be chosen uniformly over k. Thus, when ε > 0 is sufficiently small, Lemma lm:iterNC

allows us to apply Proposition

pr:apla

and Remark

rk:mu1half 2 after it to the measure η k for each k = 1, . . . , s max . Thus, either

η k P δ 2 2 ≤ δ κ/2 or η k+1 P δ 2 2 ≤ µ k+1 P δ 2 2 ≤ δ ε1 η k P δ 2 2 .
We deduce that there exists s ∈ {1, . . . , s max } such that

η s P δ 2 2 ≤ δ κ/2 .
Remembering ( eq:iterNC3 2.17), we apply Lemma lm:L2Fourier 2.6 to obtain, for all ξ with ξ = δ -1 ,

| η s * η s (ξ)| ≤ δ κ/(2d+6) ≤ δ τ .
Here we assumed ε sufficiently small compared to κ/d.

For k = 1, . . . , s, apply Lemma lm:ordre

2.7 with = 1 and m = 2 k to µ s-k+1 = η * 2 s-k η * 2 s-k . We obtain η * 2 k+1 s-k (ξ) 2 2 k ≤ µ * 2 k s-k+1 (ξ)
. Moreover, by ( eq:iterNC2 2.16), µ s-k+1 differs from η s-k+1 by a measure of total mass at most

δ τ . Hence µ * 2 k s-k+1 (ξ) ≤ η * 2 k s-k+1 (ξ) + O k (δ τ ).
From the above, we deduce using a simple recurrence that for all k = 1, . . . , s, 1) . In particular, µ * 2 s 1 (ξ) s δ ετ , which allows to conclude since µ differs from µ 1 by a measure of total mass at most δ τ .

µ * 2 k s-k+1 (ξ) k δ τ /O k (

Non-concentration in subvarieties sc:noncon

Our goal here is to prove that the law of a large random matrix product satisfies some regularity conditions.

Throughout this section, unless otherwise stated, µ denotes a probability measure on SL d (Z). As in the introduction, Γ denotes the subsemigroup generated by Supp(µ), G < SL d is the Zariski closure of Γ, and G = G(R) its group of R-rational points. We also let E denote the subalgebra of M d (R) generated by G, and fix a norm on the space of all polynomial functions on E. We shall prove two nonconcentration statements for the distribution of a random matrix product. The first one shows that the law µ * n at time n of the random matrix product is not concentrated near affine subspaces of the algebra E.

W ⊂ E , ∀n ≥ 1, ∀ρ ≥ e -n , µ * n ({g ∈ Γ | d(g, W ) ≤ ρ g }) µ ρ κ .
The second result concerns general subvarieties of the algebra E, with the caveat that we have to replace µ * n by an additive convolution power of itself, to avoid some obstructions. It is also worth noting that the quantification of the non-concentration is slightly weaker than in the case of affine subspaces. Given an integer D ≥ 1 and given ω > 0, there exists c > 0 and n 0 ∈ N such that the following holds. Let f : E → R be a polynomial function of degree D. Writing f D for its degree D homogeneous part, we have

∀k ≥ dim(E), ∀n ≥ n 0 , µ k * n x ∈ E | |f (x)| ≤ e (Dλ1-ω)n f D ≤ e -cn .
These two propositions will allow us to apply the results of the previous section to the law µ * n at time n of an irreducible random walk on SL d (Z). This will yield Theorem thm:decay

below.

Non-concentration estimates for subvarieties can sometimes be obtained by some linearization techniques, as is done in aoun [START_REF] Aoun | Transience of algebraic varieties in linear groups-applications to generic Zariski density[END_REF]. But this approach does not seem to yield a uniform statement for subvarieties of bounded degree, which is crucial for our application.

The argument developed in this section relies on the spectral gap property modulo primes for finitely generated subgroups of H(Z), where H is a semisimple Qsubgroup of SL n , a still rather recent result obtained by Salehi Golsefidy and Varjú SGV [START_REF] Golsefidy | Expansion in perfect groups[END_REF] after several important works in this direction, starting with Helfgott 3.1. Prelude : Expansion in semisimple groups. Since elements in Γ have integer coefficients, G is defined over Q, so we may choose a set of defining polynomials with coefficients in Z. Given a prime number p, this allows to consider the variety G p defined over F p by the reduction modulo p of the polynomials defining G.

On the space 2 (G p (F p )) of square-integrable functions on G p (F p ), we shall consider the convolution operator

T µ : 2 (G p (F p )) → 2 (G p (F p )) f → µ * f Let 2 0 (G p (F p )) ⊂ 2 (G p (F p ))
denote the subspace of functions on G p (F p ) having zero mean.

The theorem we shall need is the following; up to some minor modifications, it appears in Salehi Golsefidy and Varjú Proof. For any finite subset S ⊂ Γ, denote by Z e (S) the identity component of the Zariski closure of the semigroup generated by S. Let S 0 ⊂ Γ be a finite subset such that H := Z e (S 0 ) has maximal dimension among these subgroups. Then H is also maximal for the order of inclusion, because Z e (S 0 ) ⊂ Z e (S 0 ∪ S) are both irreducible subvarieties and have the same dimension.

In particular, for any γ ∈ Γ, Moreover, for any γ ∈ Γ, there is k ≥ 1 such that γ k ∈ Z e ({γ}) ⊂ H. Hence the image π(Γ) of Γ in G/H is a torsion group. By the Jordan-Schur theorem raghunathan [START_REF] Raghunathan | Discrete subgroups of Lie groups[END_REF]Theorem 8.31], π(Γ) is virtually abelian. But π(Γ) is Zariski dense in G/H. Hence the Zariski closure of any of its subgroups of finite index contains the identity component of G/H. Therefore, the identity component of G/H is both semisimple and abelian, hence trivial. It follows that G/H is finite. Thus, by adding a finite number of elements to S 0 , we can make sure that S 0 generates a Zariski dense subsemigroup in G.

γHγ -1 = Z e (γS 0 γ -1 ) ⊂ H. Hence H is a normal subgroup in G, since Γ is Zariski dense. By
lm:S-kSk Lemma 3.5. Let G be a connected semisimple algebraic group defined over Q. Let S ⊂ G(Q) be a finite subset which generates a Zariski dense subgroup. Then there exists k ≥ 1 such that the symmetric set S -k S k also generates a Zariski dense subgroup in G. On the other hand, there exists a neighborhood U of the identity in G(Q p ) and δ > 0, such that if H is an algebraic subgroup of G, then H(Q p ) is not δ-dense in U . Indeed, the Lie algebra h of H(Q p ) satisfies dim Qp h < dim Qp g, and we can distinguish two cases. If the normalizer N (h) of h in G(Q p ) does not contain an open subgroup, then we conclude using saxce_producttheorem [18, Lemma 2.2], which is still valid over Q p . Otherwise, h is an ideal in g, so that H is a sum of simple factors of G; there are only finitely many such groups.

Proof. By a result of Nori

Proof of Theorem

T µ S 2 0 (Gp(Fp)) ≤ 1 -c,
where µ S denotes the normalized counting measure on S. Then, we can write

μ * k * µ * k = αµ S + (1 -α)µ
where α > 0 and µ is some probability measure on SL d (Z). Thus, for any prime number p sufficiently large, using the fact that

T * µ = T μ, T k µ 2 2 0 (Gp(Fp)) ≤ (T * µ ) k T k µ 2 0 (Gp(Fp)) = T μ * k * µ * k 2 0 (Gp(Fp)) ≤ α T µ S 2 0 (Gp(Fp)) + (1 -α) T µ 2 0 (Gp(Fp)) ≤ 1 -αc.
One can also interpret Theorem thm:specgap 3.3 as a statement on the speed of equidistribution of the random walks associated to µ on the Cayley graphs G p (F p ). This is explained for instance in HLW [26, §3.1] for the case of simple random walks on a family of expander graphs. In our setting, we obtain the following corollary, whose proof is left to the reader. For a prime number p, we denote by π p : Z → F p the reduction modulo p. By abuse of notation we extend the domain of definition of π p to any free Z-module. cr:gapEquid Corollary 3.6. Let d ≥ 2 and let µ be a probability measure on SL d (Z) such that the Zariski closed subgroup G generated by µ is semisimple. There exists C ≥ 0 such that for every prime number p sufficiently large, and n ≥ C log p , for all a ∈ M d (F p ),

µ * n ({g ∈ M d (Z) | π p (g) = a}) ≤ 2 |G p (F p )| .
We conclude this paragraph by a lemma that will allow us to use the spectral gap theorem in our setting of random walks on the torus. Proof. Since G acts irreducibly on a finite-dimensional vector space, it is a reductive group, and can be written as an almost product

G = Z • S,
where Z is a torus, central in G, and S is semisimple, with Z ∩ S finite. The group Z is equal to the intersection of G with the center of the algebra E generated by G. Since E is a simple algebra over R, its center can be identified with R or C. Note that the restriction of the determinant on M d (R) to the center of E is simply a power of the usual norm on C, and since G ⊂ SL d (R), the group Z must be included in the group of complex numbers of norm 1. In particular, G/S Z/Z ∩ S is compact. Now since G is defined over Q, the projection map G → G/S is given by some polynomial map with rational coefficients. In particular, the image F of G ∩ SL d (Z) inside Z/Z ∩S is made of matrices whose entries are rational with bounded denominators, and is therefore finite. But G equals the Zariski closure of G∩SL d (Z), so the image of G itself under the projection G → G/S is finite, i.e. G is semisimple.

3.2.

Escaping from subvarieties: a consequence of the spectral gap. The aim of this subsection is to establish the following proposition, using the results of the previous paragraph. There exists a constant c > 0 depending on µ such that for all polynomials f ∈ Q[M d ], of degree at most D and not vanishing on G, we have

∀n ≥ 1, µ * n ({g ∈ Γ | f (g) = 0}) µ,D e -cn .
Indeed, this is a general version of BourgainGamburd_modpnII [START_REF] Bourgain | Expansion and random walks in SL d (Z/p n Z). II[END_REF]Corollary 1.1] which is stated for the group SL d and for the simple random walk on the Cayley graph. The proof is essentially the same. But since it will be important for us to know that the upper bound depends only on the degree of f , we provide a detailed argument.

We need the following lemma, which is a consequence of the Lang-Weil inequality. It will be important for us to have an estimate which is uniform for subvarieties of bounded complexity. lm:LWcplxity Lemma 3.9. Given a geometrically irreducible subvariety V ⊂ A d defined over Q and an integer D ≥ 1, there exists p 0 = p 0 (V, D) such that the following holds. Let f ∈ Q[X 1 , . . . , X d ] be a polynomial of degree at most D. Assume that f does not vanish on V . Then for every prime number p ≥ p 0 ,

|{π p (x) ∈ F d p | x ∈ V ∩ Z d , f (x) = 0}| V,D p -1 |V p (F p )| where V p denotes the reduction modulo p of V .
Note that to speak about reductions modulo p of a variety over Q, we need to choose a model over Z. But since V is a subvariety of an affine space, among such choices, there is a canonical one. See the first paragraph in the proof below. V p is geometrically irreducible for p ≥ p 0 , where p 0 is a constant depending only on the embedding V ⊂ A d .

Proof. We abbreviate Q[X 1 , . . . , X d ] as Q[X] and Z[X 1 , . . . , X d ] as Z[X]. Let I ⊂ Q[X]
On the one hand applying the Lang-Weil inequality LangWeil [27, Theorem 1] to the irreducible variety V p , we obtain eq:LWlower eq:LWlower

(3.1) |V p (F p )| ≥ 1 2 p dim(Vp) .
On the other hand, one can prove, using Gröbner bases CoxLittleOShea [17, Chapter 2], that given an integer D ≥ 1 there is p 0 = p 0 (V, D) such that for all p ≥ p 0 and all

f ∈ Q[X] \ I of degree at most D there is h ∈ (Qf ⊕ I) ∩ Z[X] of degree at most O V,D (1) and such that π p (h) / ∈ π p (I Z ). For such h, we have {x ∈ V ∩ Z d | f (x) = 0} ⊂ {x ∈ V ∩ Z d | h(x) = 0}
and hence

|{π p (x) ∈ F d p | x ∈ V ∩ Z d , f (x) = 0}| ≤ |{x ∈ V p (F p ) | π p (h)(x) = 0}|.
The right-hand side is the number of F p -points in the subvariety V p ∩ {π p (h) = 0}. This subvariety has dimension at most dim(V p ) -1 since V p is irreducible and π p (h) / ∈ π p (I Z ). Thus, applying a version of the Schwarz-Zippel estimate, like LangWeil [27, Lemma 1], and using the fact that the complexity controls the degree, we get

(3.2) |(V p ∩ {π p (h) = 0})(F p )| V,D p dim(Vp)-1 .
Together with ( eq:LWlower

|{π p (g) ∈ M d (F p ) | g ∈ Γ, f (g) = 0}| G,D p -1 |G p (F p )|
for every prime number p ≥ p 0 (G, D). Combined with Corollary cr:gapEquid 3.6, this yields

µ * n ({g ∈ Γ | f (g) = 0}) G,D p -1 .
for all n ≥ C log p, where C is a constant depending only on µ. We conclude by choosing p to be a prime number such that p e n/C and p ≥ p 0 .

3.3. Interlude : large deviation estimates for random matrix products.

In this subsection, µ is a Borel probability measure on SL d (R), not necessarily supported on matrices with integer coefficients. By Γ we denote the closure of the subsemigroup generated by Supp(µ) and by G the group of R-points of the Zariski closure of Γ.

Let us first recall the large deviation estimates for random matrix products. This result is originally due to Lepage LePage [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF], and the version below is taken from Bougerol

BougerolLacroix [6, Theorem V.6.2]. For g ∈ GL d (R), denote by σ 1 (g) ≥ • • • ≥ σ d (g) > 0
the singular values of g ordered decreasingly.

thm:LargeD Theorem 3.10 (Large deviation estimates). Let µ be a Borel probability measure on GL d (R) having a finite exponential moment. Let λ k denote the k-th Lyapunov exponent associated to µ. Assume that G acts strongly irreducibly on R d . For any ω > 0, there is c > 0, n 0 > 0 such that the following holds.

it:LargeDn (i) For all n ≥ n 0 , µ * n g ∈ Γ | | 1 n log g -λ 1 | ≥ ω ≤ e -cn .
it:LargeDsv (ii) For all k = 1, . . . , d and all n ≥ n 0 ,

µ * n g ∈ Γ | | 1 n log σ k (g) -λ k | ≥ ω ≤ e -cn . it:LargeDnc (iii) For all n ≥ n 0 , For all v ∈ R d \ {0}, µ * n g ∈ Γ | 1 n log gv v -λ 1 ≥ ω ≤ e -cn .

Item

it:LargeDn (i) is, of course, a special case of Item it:LargeDsv (ii) since σ 1 (g) = g . One consequence of the above theorem that will be useful to us is the following proposition.

pr:LDrhokappa Proposition 3.11. Let µ be a Borel probability measure on GL d (R) having a finite exponential moment. Assume that G acts strongly irreducibly on R d . Then there exists κ > 0 such that for all n ≥ 1 and all ρ ≥ e -n , for every v ∈ R d \ {0}, eq:normgv eq:normgv

µ * n g ∈ Γ | gv ≤ ρ g v µ ρ κ . Proof. Let r denote the proximal dimension of G, i.e. r = min{rank g ; g ∈ RG \ {0}}.
(3.3) ∀g ∈ GL d (R), ∀v ∈ R d \ {0}, gv ≥ d (Rv, W - g ) g v ,
where d is defined, for any two subspaces V, W ⊂ R d of R d with respective orthonormal bases (v 1 , . . . , v s ) and (w 1 , . . . , w t ), by the formula

d (V, W ) = v 1 ∧ • • • ∧ v s ∧ w 1 ∧ • • • ∧ w t .
Then, BFLM [9, Lemma 4.5] applied to the transposed random walk, which is also proximal, shows that there exists κ > 0 such that eq:V+rhokappa eq:V+rhokappa

(3.4) ∀ρ ≥ e -n , µ * n g ∈ Γ | d (V + t g , v ⊥ ) ≤ ρ µ ρ κ . Noting that (W - g ) ⊥ = V + t g and that d (V, W ) = d (V ⊥ , W ⊥ ) if dim V +dim W = d,
of ∧ r R d into G-invariant subspaces ∧ r R d = Λ + ⊕ Λ 0
such that the action of G on Λ + is strongly irreducible and proximal and moreover, eq:gLambda+ eq:gLambda+

(3.5) ∀g ∈ G, (∧ r g) |Λ+ G g r ≥ ∧ r g .
Denote by π + : ∧ r R d → Λ + the projection with respect to this decomposition. By

Breuillard [START_REF] Breuillard | A non concentration estimate for random matrix products[END_REF]Lemma 3.3], for any v ∈ R d , we can find a subspace P ⊂ R d of dimension r and containing v such that eq:pickPfromv eq:pickPfromv

(3.6) π + (v P ) G v P ,
where v P ∈ ∧ r R d is the wedge product of the elements of a basis of P . Now, observe that ( eq:normgv 3.3) still holds without the proximality assumption, and we have, for every g in GL d (R), d (Rv, W - g ) ≥ d (P, W - g ). Hence, it suffices to prove, for some κ > 0 and n 0 ≥ 1, eq:W-rhokappa eq:W-rhokappa

(3.7) ∀n ≥ n 0 , ∀ρ ≥ e -n , µ * n g ∈ Γ | d (P, W - g ) ≤ ρ µ ρ κ . By Breuillard [15, Lemma 4.2], ∀g ∈ GL d (R), (∧ r g)v P ∧ r g v P ≤ d (P, W - g ) + σ r+1 (g) σ r (g) .
Combined with ( eq:gLambda+

3.5) and (

eq:pickPfromv

3.6), this yields ∀g ∈ G, (∧ r g) |Λ+ π + (v P ) (∧ r g) |Λ+ π + (v P ) G d (P, W - g ) + σ r+1 (g) σ r (g) .

By a result of Guivarc'h-Raugi

GuivarchRaugi [START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF], λ r > λ r+1 . Applying Theorem thm:LargeD

3.10

it:LargeDsv (ii) to k = r and r + 1 and ω 0 = (λ r -λ r+1 )/3 > 0, we get c > 0 and n 0 ≥ 1 such that

∀n ≥ n 0 , µ * n g ∈ Γ | σ r+1 (g) σ r (g) ≤ e -ω0n ≥ 1 -e -cn .
Note that e -cn ≤ ρ c . The desired estimate ( eq:W-rhokappa 3.7) then follows by the proximal case applied to the induced random walk on Λ + and to the vector π + (v P ) ∈ Λ + . Given an integer D ≥ 1, there exist constants C > 0, c > 0 and n 0 ≥ 1 depending on µ and D such that

∀f ∈ R[G] ≤D , ∀n ≥ n 0 , µ * n g ∈ Γ | |f (g)| < e -Cn f ≤ e -cn .
Proof. By Theorem thm:LargeD

3.10

it:LargeDn (i), there is c > 0 such that for n large enough

µ * n g ∈ Γ | g ≥ e 2λ1n ≤ e -cn ,
where λ 1 is the top Lyapunov exponent associated to the random walk defined by µ on R d . Thus, we are left to bound from above the µ * n -measure of the set

A C = g ∈ Γ | |f (g)| ≤ e -Cn f and g ≤ e 2λ1n .
Let us abbreviate

V = R[G] ≤D . Let V Q denote the set of functions f ∈ V which are Q-rational, i.e. represented by polynomials on M d with coefficients in Q. Note that V Q defines a Q-structure on V .
We claim that if C is chosen large enough then A C must be contained in some subvariety {f 0 = 0} where f 0 ∈ V Q \ {0}. Then Proposition pr:gapEscape 3.8 applied to f 0 allows to conclude.

For every g ∈ Γ let ev g : V → R be the evaluation map

∀v ∈ V, ev g (v) = v(g).
Since the matrices g ∈ A C have integer coefficients, the intersection

W = g∈A C ker(ev g ) is a subspace of V defined over Q, i.e. W = R ⊗ Q (W ∩ V Q ).
We want to show that W ∩ V Q contains nonzero element. Assume for a contradiction that W = {0}. Write s = dim(V ). We can choose g 1 , . . . , g s ∈ A C such that

{0} = s i=1
ker(ev gi ).

Fix a basis (v 1 , . . . , v s ) of V in which each element is represented by a polynomial on M d with coefficients in Z. Thus, the map Φ :

V → R s defined by ∀f ∈ V, Φ(v) = ev gi (v) i
is invertible and has integer coefficients when expressed in the basis (v 1 , . . . , v s ) and the standard basis of R s . Thus, in these bases, the determinant of

Φ satisfies |det(Φ)| ≥ 1. Moreover, Φ max 1≤i,j≤s |ev gi (v j )| G,D max 1≤i≤s g i D ≤ e 2Dλ1n .
It follows that

Φ -1 Φ s-1 |det(Φ)| ≤ e 2Dsλ1n .
For f ∈ V , by definition of A C , we have

Φ(f ) = max 1≤i≤s |f (g i )| ≤ e -Cn f . Thus, f ≤ Φ -1 Φ(f ) G,D e (2Dsλ1-C)n f . We get a contradiction if C is chosen to be larger than 2Dsλ 1 + O G,D (1).
The following is a variant and an easy consequence of the previous lemma. Given an integer D, there exist constants C > 0, c > 0 and n 0 ≥ 1 depending on µ and D such that

∀f ∈ R[G] ≤D , ∀n ≥ n 0 , µ * n g ∈ Γ | |f (g)| < e -Cn g f ≤ e -cn .
Proof. For any n ≥ 1 and any C > 0, we have

µ * n g ∈ Γ | |f (g)| < e -(C+2λ1)n g f ≤ µ * n g ∈ Γ | |f (g)| < e -Cn f +µ * n g ∈ Γ | g ≥ e 2λ1n .
We conclude by using Lemma lm:mundef<e 3.12 for the first term and Theorem thm:LargeD

3.10

it:LargeDn (i) for the second.

3.5. Non-concentration near affine subspaces. We now want to prove Proposition pr:NCaffine 3.1. Of course, if we are to show that the random walk does not concentrate near any proper affine subspace in the algebra E, we should first check that the group G is not trapped in any proper affine subspace. Proof. Equivalently, we have to show that the linear span W = Span(G -1) of G -1 is E. For this, it suffices to prove that 1 ∈ W . Firstly, W is closed under multiplication. Indeed, any product between two elements of W is a linear combination of elements of the form (g -1)(h -1) with g, h ∈ G and we have

(g -1)(h -1) = (gh -1) -(g -1) -(h -1) ∈ W.
Secondly, any subspace of R d preserved by W is preserved by G, hence the only subspaces preserved by W are R d and {0}. We conclude by using the following algebraic lemma.

Momentarily, in the next lemma and its proof, algebras are not assumed to be unital. Thus, an subalgebra is a linear subspace that is closed under multiplication. Accordingly, a left (resp. right) ideal, is a subspace preserved under multiplication on the left (resp. right) by all elements of the algebra. Proof. Let W ⊂ M d (R) be such a subalgebra. We first show that the only nilpotent right ideal of W is the zero ideal. Indeed, let I be a nonzero nilpotent right ideal of W . Let k be the maximal number such that

I k = 0. Let f 0 ∈ I k . Since I k+1 = 0, we have f 0 (R d ) ⊂ f ∈I ker(f ).
The intersection on the right-hand side is preserved by W and not equal to R d , because I is a nonzero right ideal. Then f 0 must be zero, which contradicts I k = 0. Thus, W is an algebra without radical

W . Its image 1 W (R d ) is preserved by W and nonzero since W is nonzero. Hence 1 W (R d ) = R d and 1 W ∈ GL d (R). Then 1 2 W = 1 W forces 1 W to be the identity of M d (R).
2 More precisely, we apply the theorem to algebras, or rings with operator domain R, using the terminology of Van

Proof of Proposition

pr:NCaffine

By Lemma

lm:notaffine 3.14, R[G] ≤1 is isomorphic to R ⊕ E * , the space of affine mappings from E to R. On R ⊕ E * , let G act by ∀g ∈ G, ∀f ∈ R ⊕ E * , ∀x ∈ E, (g • f )(x) = f (xg).
By Lemma lm:mundef<eg 3.13, there exist C 1 ≥ 1 and c 1 > 0 such that eq:mu*mdef eq:mu*mdef

(3.8) ∀f ∈ R ⊕ E * , ∀m ≥ 1, µ * m g ∈ Γ | |f (g)| < e -C1m g f µ e -c1m . Given a proper affine subspace W ⊂ E, there exists f ∈ R ⊕ E * such that its linear part f 1 ∈ E * has norm f 1 = 1 and ∀g ∈ E, d(g, W ) = |f (g)|.
Let ρ ≥ e -n . Pick m such that e -C1m ρ 1/2 . Using the relation µ * n = µ * m * µ * (n-m) , we have

µ * n ({g ∈ Γ | |f (g)| ≤ ρ g }) = Γ µ * m ({g ∈ Γ | |(h • f )(g)| ≤ ρ gh }) dµ * (n-m) (h)
We distinguish two cases according to whether ρ h ≤ e -C1m h • f . If this is the case, then ρ gh ≤ e -C1m g h • f and then by ( eq:mu*mdef

3.8), µ * m g ∈ Γ | |(h • f )(g)| ≤ ρ gh µ e -c1m ρ c1/2C1 . Otherwise, h • f ≤ ρe C1m h ≤ ρ 1/2
h by the choice of m. Thus,

µ * n ({g ∈ Γ | |f (g)| ≤ ρ g }) µ ρ c1/2C1 + µ * (n-m) h ∈ Γ | h • f ≤ ρ 1/2 h . Now observe that E * is
f 1 = 1, we obtain π i (f ) G 1 for some i. Then h • f ≤ ρ 1/2 h implies hπ i (f ) G ρ 1/2 h π i (f ) . Hence µ * (n-m) h ∈ Γ | h • f ≤ ρ 1/2 h ≤ µ * (n-m) h ∈ Γ | hπ i (f ) ≤ C 2 ρ 1/2 h π i (f )
where C 2 is a constant depending only on G. By our choice of m, we have C 2 ρ 1/2 ≥ e -(n-m) . Hence, by Proposition

pr:LDrhokappa 3.11, µ * (n-m) h ∈ Γ | hπ i (f ) ≤ C 2 ρ 1/2 h π i (f ) µ ρ κ2/2
where κ 2 > 0 is the constant given by Proposition pr:LDrhokappa

which depends only on µ.

This proves the desired estimate with κ = min{ c1 2C1 , κ2 2 }. 3.6. Escaping a larger neighborhood of a subvariety. The rest of this section is devoted to the proof of Proposition pr:NCdet 3.2. The idea is to generalize what we did above for affine subspaces. This time, the variety that we want to avoid is defined by a general polynomial map f on the algebra E, so that we shall have to consider the representation ρ :

G → GL(R[G]) defined by ∀g ∈ G, ∀f ∈ R[G], ∀x ∈ G, (ρ(g)f )(x) = f (xg).
We refer to finite-dimensional subrepresentations of this representation as G-modules. For a G-module M , we denote by λ 1 (µ, M ) the top Lyapunov exponent associated to the random walk on M defined by µ:

λ 1 (µ, M ) = lim 1 n G log ρ(g) |M dµ * n (g)
where denotes some operator norm. For a real number λ ≥ 0, define M λ to be the sum of submodules M of R[G] ≤D such that λ 1 (µ, M ) ≥ λ. Let p λ : R[G] ≤D → M λ be an epimorphism of G-modules onto M λ . Remark that M λ is a sum of isotypical components in R[G] ≤D so that p λ is uniquely defined. Given D, λ ≥ 0 and ω > 0, there is c > 0 and n 0 ≥ 1 (depending also on µ)

such that ∀f ∈ R[G] ≤D , ∀n ≥ n 0 , µ * n g ∈ Γ | |f (g)| ≤ e (λ-ω)n p λ (f ) ≤ e -cn .
Proof. Note that, for any 0 < m < n, we have eq:n=n-m+m eq:n=n-m+m

(3.9) µ * n g ∈ Γ | |f (g)| ≤ e (λ-ω)n p λ (f ) = Γ µ * m g ∈ Γ | |(ρ(h)f )(g)| ≤ e (λ-ω)n p λ (f ) dµ * (n-m) (h) For any f ∈ R[G] ≤D , there is a simple G-module M contained in M λ such that p M (f ) G,D,λ p λ (f ) where p M : M λ → M is a projection of G-modules.
By definition of M λ , the top Lyapunov exponent in M satisfies λ 1 (µ, M ) ≥ λ. Note that since G is Zariski connected, the G-action on the simple module M is strongly irreducible. Hence, by the large deviation estimate Theorem thm:LargeD

3.10

it:LargeDnc (iii), there is c > 0 and n 0 ≥ 1 such that for all m > 0 satisfying n -m ≥ n 0 ,

µ * (n-m) h ∈ Γ | ρ(h)p M (f ) ≤ e (λ-ω 2 )(n-m) p M (f ) ≤ e -c(n-m) ,
and hence eq:LDPrhohf eq:LDPrhohf

(3.10) µ * (n-m) h ∈ Γ | ρ(h)f ≤ e (λ-ω 2 )(n-m) p λ (f ) ≤ e -c(n-m) .
Applying Lemma lm:mundef<e 3.12 to the function ρ(h)f , for h ∈ Γ, we obtain ∀m ≥ m 0 eq:mumdef eq:mumdef

(3.11) µ * m g ∈ Γ | |(ρ(h)f )(g)| < e -Cm ρ(h)f ≤ e -cm ,
for some C > 0, c > 0 and m 0 ≥ 1. Setting m = ωn 2(C+λ) , so that

(λ -ω)n -(λ - ω 2 )(n -m) ≤ -Cm,
the desired inequality follows from ( eq:n=n-m+m 3.9), ( eq:LDPrhohf 3.16, we need to be able to say when a regular function has a nonzero component in a simple submodule of large Lyapunov exponent. lm:criterionL Lemma 3.17. Let µ be a probability measure on SL d (R), d ≥ 2, with some finite exponential moment. Assume that the group Γ generated by µ is non-compact and acts irreducibly on R d , and that its Zariski closure G is connected.

Let f ∈ R[M d ] be a polynomial of degree D ≥ 1 whose degree D homogeneous part does not vanish on the algebra E generated by G. The following holds for every integer k ≥ dim(E). Consider the polynomial

F ∈ R[M k d ] defined by F (x 1 , . . . , x k ) = f (x 1 + • • • + x k ) and let F ∈ R[G k ] be the restriction of F to G k . Then p(F ) = 0 where p : R[G k ] ≤D → R[G k ] ≤D is the projection to the sum of all simple G k - submodules M of R[G k ] ≤D having λ 1 (µ ⊗k , M ) ≥ Dλ 1 (µ, R d ).
We shall use the theory of the highest weight as well as the theory of random walks on semisimple groups. So let us fix some notation and recall briefly the needed results. Let g denote the Lie algebra of G. Let K be a maximal compact subgroup of G. Inside the orthogonal complement, with respect to the Killing form, of the Lie algebra of K, we choose a Cartan subspace a of g. Every algebraic representation of G is diagonalizable for a. That is, for every G-module M , we have

M = χ∈a M χ
where for each χ ∈ a * , M χ is the associated weight space

M χ = {v ∈ M | ∀a ∈ a, exp(a) • v = e χ(a) v}.
The linear forms χ ∈ a * for which M χ = {0} are called the weights of M . Denote by Σ(M ) the sets of weights of M .

The set of of nontrivial weights of the adjoint representation of G is the set of restricted roots. We denote it by Σ. It forms a root system. We fix a set Σ + of positive roots and denote by a + the associated Weyl chamber:

a + = {a ∈ a | ∀α ∈ Σ + , α(a) ≥ 0}.
We also write a ++ to denote the interior of the Weyl chamber:

a ++ = {a ∈ a | ∀α ∈ Σ + , α(a) > 0}.
Let g ∈ G. The Cartan projection κ(g) of g is the a + -part in its Cartan decomposition, that is, the unique element in a + such that g ∈ K exp(κ(g))K. The law of large numbers for a semisimple group, BenoistQuint [4, Theorem 10.9], says that there is an element λ(µ) in a ++ , called the Lyapunov vector associated to µ, such that

λ(µ) = lim n→+∞ 1 n G κ(g) dµ * n (g).
If M is a simple G module, then M has a highest weight, denoted by χ M ∈ Σ(M ), so that for any weight χ ∈ Σ(M ), χ M -χ is a sum of positive roots. By BenoistQuint [4, Corollary 10.12], we have

λ 1 (µ, M ) = χ M ( λ(µ)).
Now let us recall the definition of the limit set of the group G in M d (R). We write RG for the set of all elements M d (R) of the form λg with λ ∈ R, and g ∈ G.

Let RG denote the closure of RG in M d (R) for the norm topology. Let r G denote the proximal dimension of G, defined by eq:defproxdim eq:defproxdim (3.12) r G = min rank(π) | π ∈ RG \ {0} .

The limit set of G in M d (R) is defined to be

Π G = π ∈ RG | rank(π) = r G . lm:limitset Lemma 3.18. Let G < SL d be connected semisimple R-group. Assume that G = G(R) acts irreducibly on R d . Let π 0 ∈ M d (R)
be the spectral projector to the weight space associated to the highest weight. Then

Π G = R * Kπ 0 K = R * Gπ 0 G.
If moreover G is not compact then, writing E = Span R (G), the sum-set

Gπ 0 G + • • • + Gπ 0 G dim E times
contains an open subset of E.

Proof. Let χ 0 = χ R d ∈ Σ(R d ) denote the highest weight of the simple G-module R d . For a weight χ ∈ Σ(R d )
, let π χ be the spectral projector to the associated weight space. Let a ∈ a ++ be any element. We have

exp(na) -1 exp(na) = χ∈Σ(R d )
e n(χ(a)-χ0(a)) π χ .

Now by definition of the highest weight, χ(a) -χ 0 (a) < 0 for χ = χ 0 . It follows that

π 0 = lim n→+∞ exp(na) -1 exp(na) ∈ RG.
Let π ∈ RG be another nonzero element. There exists sequences

(λ n ) ∈ R N and (g n ) ∈ G N such that π = lim n→+∞ λ n g n . Let g n = k n exp(a n ) n ∈ K exp(a +
)K be the Cartan decomposition of g n . By compactness of K, replacing (g n ) by a subsequence if necessary, we may assume that k n converges to k ∈ K and n converges to . Then

k -1 π -1 = lim n→+∞ λ n exp(a n ) Observe that exp(a n ) = χ∈Σ(R d )
e χ(an) π χ .

Hence λ n exp(a n ) = λ n e χ0(an) (π 0 + χ∈Σ(R d )\{χ0}
e χ(an)-χ0(an) π χ ).

Note that e χ(an)-χ0(an) ≤ 1 for all χ ∈ Σ(R d ) and n ≥ 1. We deduce that λ n e χ0(an) converges to λ = 0, for otherwise π would be zero. Moreover,

rank(π) = rank(k -1 π -1 ) ≥ rank(π 0 ).
Equality holds if and only if lim n→+∞ e χ(an)-χ0(an) = 0 for all χ ∈ Σ(R d ) \ {χ 0 }, which in turn is equivalent to

k -1 π -1 = λπ 0 .
Therefore, r G = rank(π 0 ) and Π G ⊂ R * Kπ 0 K. We conclude by noticing that Π G is invariant under multiplication by G on both sides :

R * Kπ 0 K ⊂ R * Gπ 0 G ⊂ Π G .
For the last assertion, assume that G is not compact. Then, χ 0 = 0 and therefore χ 0 (a) = R. For a ∈ a, exp(a)π 0 = e χ0(a) π 0 , so that R * + π 0 ⊂ Gπ 0 and hence

R * + Gπ 0 G ⊂ Gπ 0 G. Since the action of G on R d is irreducible, E is an simple algebra over R,

by a version of Wedderburn's theorem

VanderWaerden [35, 2. page 194]. Observe that Span R (Gπ 0 G) is a nontrivial two-sided ideal of E, hence Span R (Gπ 0 G) = E. Therefore we can pick dim(E) elements (π 1 , . . . , π dim(E) ) from Gπ 0 G making a basis of E. We conclude that

Gπ 0 G + • • • + Gπ 0 G dim E times ⊃ R * + π 1 + • • • + R * + π dim(E)
contains an open subset of E.

Proof of Lemma

lm:criterionL 3.17. The Lie algebra of G k is g ⊕ • • • ⊕ g, in which we choose b = a ⊕ • • • ⊕ a to be the Cartan subspace. Then the associated restricted root system is the direct sum

Σ • • • Σ ⊂ b * . We choose Σ + • • • Σ + as the set of positive roots so that b + = a + × • • • × a + is the corresponding Weyl chamber and λ(µ ⊗k ) = ( λ(µ), . . . , λ(µ)) ∈ b +
is the Lyapunov vector associated to the random walk defined by µ ⊗k .

For any algebraic representation π of G k , we denote by Σ(G k , π) the set of weights of π with respect to b.

Let σ : G → GL(R d ) denote the standard representation of G and, for i = 1, . . . , k, let σ i : G k → G → GL(R d ) denote the representation of G k obtained by composing the projection G k → G to the i-th factor with σ. Note that for each i, there is a natural bijection between Σ(G k , σ i ) → Σ(σ), χ → χ such that the weight χ is the composition of the i-th projection with χ ∈ Σ(σ).

Let

G k act on R[M k d ] ≤D by right translation. Let ρ : G k → GL(R[M k d ] ≤D ) denote the corresponding representation. Then, ρ is equivalent to D j=0 Sym j σ 1 ⊕ • • • ⊕ σ 1 d times ⊕ • • • ⊕ σ k ⊕ • • • ⊕ σ k d times .
It follows that any weight χ ∈ Σ(G k , ρ) in ρ is the sum of at most D elements from

k i=1 Σ(G k , σ i ). In particular, λ 1 (µ ⊗k , R[M k d ] ≤D ) = max χ∈Σ(G k ,ρ) χ( λ(µ ⊗k )) ≤ D max i max χ∈Σ(G k ,σi) χ( λ(µ ⊗k )) = D max i max χ∈Σ(G k ,σi) χ( λ(µ)) ≤ Dλ 1 .
Since G is not compact, λ 1 is positive, by a result of Furstenberg Furstenberg1963 [START_REF] Furstenberg | Noncommuting random products[END_REF], and it follows that

λ 1 (µ ⊗k , R[M k d ] <D ) < Dλ 1 . The G k -module R[G k ] <D is a quotient of R[M k d ] <D ,

whence

eq:l1of<Dpart eq:l1of<Dpart

(3.13) λ 1 (µ ⊗k , R[G k ] <D ) < Dλ 1 . Let f D ∈ R[M d ] denote the degree D homogeneous part of f . Then FD ∈ R[M k d ] defined by ∀x 1 , . . . , x k ∈ M d , FD (x 1 , . . . , x k ) = f D (x 1 + • • • + x k )
is the degree D homogeneous part of F . Let F D ∈ be the restriction of FD to G. By ( eq:l1of<Dpart

3.13) and the fact that

F -F D ∈ R[G k ] <D , we get p(F ) = p(F D ).
We may therefore assume that f is homogeneous of degree D.

By Lemma lm:limitset

3.18, f does not vanish on Gπ

0 G + • • • + Gπ 0 G where π 0 = lim n→+∞ exp na exp na ∈ M d (R) for some a ∈ a ++ . Fix g = (g 1 , . . . , g k ) ∈ G k and h = (h 1 , . . . , h k ) ∈ G k such that f (g 1 π 0 h 1 + . . . g k π 0 h k ) = 0.
Writing b = (a, . . . , a) ∈ b, we have, by the homogeneity of F ,

lim n→+∞ F (g exp(nb)h) exp(na) D = lim n→+∞ F g exp(nb)h exp(na) = f (g 1 π 0 h 1 + . . . g k π 0 h k ) = 0, whence |F (g exp(nb)h)| exp(na) D .
In the rest of the proof, the implied constants in the Vinogradov notation are independent of n but might depend on other quantities, like g or h. On the one hand, exp(na) e nχσ(a) , and on the other hand,

|F (g exp(nb)h)| = ρ(exp(nb))ρ(h)F (g)) ρ(exp(nb))ρ(h)F .
Decompose R[G k ] ≤D = j M j into simple G k -modules and decompose ρ(h)F = j F j accordingly. Denote by χ Mj the highest weight of M j . Then

ρ(exp(nb))ρ(h)F ≤ j e nχ M j (b) F j .
From the previous inequalities, there must exist j such that F j = 0 and e Dnχσ(a) e nχ M j (b) F j .

Hence We have seen that χ Mj is the sum of D elements from

k i=1 Σ(G k , σ i ): χ Mj = χ 1 + • • • + χ D . Then χ Mj (b) = χ 1 (b) + • • • + χ D (b) = χ1 (a) + • • • + χD (a).
Thus we have simultaneously

Dχ σ (a) ≤ χ1 (a) + • • • + χD (a) and χ1 + • • • + χD ≤ Dχ σ
for the order over the set of weights. Since a ∈ a ++ , , this forces

χ1 = • • • = χD = χ σ .
Therefore

λ 1 (µ ⊗k , M j ) = χ Mj ( λ(µ ⊗k )) = ( χ1 + • • • + χD )( λ(µ)) = Dχ σ ( λ(µ)) = Dλ 1 .
We conclude that p(ρ(h)F ) = 0 and hence

p(F ) = ρ(h) -1 p(ρ(h)F ) = 0.
We can now easily deduce Proposition pr:NCdet

from Proposition

pr:LDPpolynome

and Lemma

lm:criterionL 3.17.

Proof of Proposition

pr:NCdet 3.2. Note that under our assumptions, G cannot be compact. Let f : E → R be a polynomial map of degree D, and denote by f D its degree D homogeneous part. Define

F ∈ R[G k ] by ∀g 1 , . . . , g k ∈ G, F (g 1 , . . . , g k ) = f (g 1 + . . . + g k ). Let p : R[G k ] ≤D → R[G k
] ≤D be the projection to the sum of all simple submodules M such that λ 1 (µ ⊗k , M ) = Dλ 1 (µ, R d ). By Lemma lm:criterionL 3.17, p(F ) = 0 implies f D = 0, and since these two expressions define seminorms on the space of polynomial maps on E, it follows that fdpf fdpf (3.14) f D p(F ) .

By Proposition

pr:LDPpolynome 3.16 applied to the random walk on G k associated to the measure µ ⊗k , with λ = Dλ 1 (µ, R d ) we get that for every ω > 0, there exists c > 0 and

n 0 ∈ N such that ∀n ≥ n 0 , µ * n g ∈ G k | |F (g)| ≤ e (λ-ω)n p(F ) ≤ e -cn .
Together with ( fdpf 3.14), this proves what we want.

3.8. Fourier decay for random walks. The relevant object here is the measure μn , obtained from µ * n after rescaling by a factor e -λ1n . This rescaling shrinks µ * n to a ball of subexponential size around 0. An important consequence of the results of this section and the previous one is the following theorem. Proof. We want to apply Theorem thm:fourier 2.1 to the measure μn , and for that, we should check that it is not concentrated near any affine subspace, nor near any translate of the set of non-invertible elements of E. This will follow from Propositions pr:NCaffine

and

pr:NCdet 3.2. Recall that given ρ > 0, we write S E (ρ) for the set

S E (ρ) = {x ∈ E | | det E (x)| ≤ ρ}.
Let D = dim(E). Under the assumptions of the theorem, we claim that there exists κ > 0 depending only µ such that for every ω > 0, there exists c = c(µ, ω) > 0 such that for every n ≥ 1, we can decompose the convolution

μ D n μ D n = η + θ
into positive Borel measures satisfying the following properties.

(i) θ(E) µ e -cn , (ii) η(E \ B E (0, e ωn )) µ e -cn , (iii) ∀x ∈ E, η(x + S E (e -ωn )) µ e -cn , (iv) ∀ρ ≥ e -n , ∀W < E affine subspace, η(W (ρ) ) µ e ωn ρ κ . To justify this claim, let η 1 be the restriction of μn to E \ B E (0, e -ωn ) and put

η = η 1 μ (D-1)

3.10

it:LargeDn (i), there is c = c(µ, ω) > 0 such that for every n ≥ 1, μn (B E (0, e -ωn )) µ e -cn and μn (E \ B E (0, e ωn )) µ e -cn .

It follows that

θ(E) µ e -cn and η E \ B E (0, 2De ωn ) µ e -cn . For x ∈ E, apply Proposition pr:NCdet 3.2 to the polynomial function y → det E (y -e λ1m x). Note that these polynomials all have degree D = dim(E) and all have the same degree D homogeneous part, namely det E . We obtain c > 0 such that

η (x + S E (e -ωn )) ≤ µ D n y ∈ E | |det E (e -λ1n y -x)| ≤ e -ωn ≤ µ D n y ∈ E | |det E (y -e λ1n x)| ≤ e (Dλ1-ω)m µ e -cn .
Since this property is preserved under additive convolution, the same holds for η. Now let W be a proper affine subspace of E. Using the definition of η 1 and Proposition pr:NCaffine 3.1, we find for every ρ ≥ e -n ,

η 1 ({g ∈ E | d(g, W ) ≤ ρ}) ≤ μn g ∈ E | d(g, W ) ≤ ρe ωn g ≤ µ n g ∈ E | d(g, W ) ≤ ρe ωn g µ e ωκn ρ κ .
Again this property is preserved under additive convolution, so that η satisfies the required conditions.

Let τ = cα 2 and ε = αω 2 . With this choice of parameters, for every n large enough, for every R ∈ [e αn 2 , e

2n α ], (i) θ(E) ≤ R -τ , (ii) η(E \ B E (0, R ε )) ≤ R -τ , (iii) for all x ∈ E, η(x + S E (R -ε )) ≤ R -τ , (iv) for all ρ ≥ R -1 and every proper affine subspace W ⊂ E, η(W (ρ) ) µ R ε ρ κ .
In other words, the assumptions of Theorem thm:fourier 2.1 are satisfied for the measure η at the scale 1/R. Therefore, for all ξ ∈ E in the range

e αn 2 ≤ ξ ≤ e 2n α , we have | η * s (ξ)| ≤ ξ -ετ .
It follows that

| (μ D n μ D n ) * s (ξ)| ≤ ξ -ετ + O s ( ξ -τ ). Applying Lemma lm:ordre 2.7 to μ D n μ D n , we get | μsn (ξ)| ≤ ξ -c0 ,
where c 0 = ετ 2(2D) s and, again, assuming that n ≥ n 0 (µ). This shows the desired result when n is a multiple of s.

The general case follows, writing n = sq + r, 0 ≤ r < s,

μn (ξ) = μsn (ξx) dμ r (x),
and using the fact that for x outside of a set of exponentially small μr -measure, There exist constants C ≥ 0, s ∈ N and τ > 0 such that the following holds. Let ν be a Borel probability measure on T d . Let t 0 ∈ (0, 1/2). Assume that for some

a 0 ∈ Z d \ {0}, | µ * n * ν(a 0 )| ≥ t 0 and n ≥ C|log t 0 |.
Then, writing N = e n(λ1-1 2s ) a 0 and M = e -τ n N , there exists a 1 M -separated set

X ⊂ T d such that ν x∈X B(x, 1 N ) ≥ t C 0 .
The proof of this statement goes by two steps. First, following BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], one applies a Fourier analytic lemma BFLM [9, Proposition 7.5] to translate the concentration of ν to a statement about its Fourier coefficients. Then, one uses the Fourier decay of μn to study the set of large Fourier coefficients at at (4.1)

A(t) = {a ∈ Z d | |ν(a)| ≥ t},
and prove the desired statement.

4.1. Concentration from the Fourier coefficients. Because it is so elementary, and yet beautiful, we include the Fourier analytic lemma needed for our argument. The reader is referred to BFLM [9, Proposition 7.5] for its ingenious proof.

Lemma 4.2. Given d ∈ N, there exists c > 0 such that if a measure ν on T d satisfies

N (A(t) ∩ B(0, N ), M ) ≥ s N M d
for some numbers s, t > 0 and some M, N ≥ 1 such that M < cN , then there exists

a 1 M -separated subset X ⊂ T d such that ν x∈X B(x, 1 N ) ≥ c(st) 3 .
Going back to the statement of Proposition pr:step1

4.1 above we see that it is enough to show that, under the same assumptions, there exist C ≥ 0, s ∈ N and τ > 0 such that, for N = e n(λ1-1 2s ) a 0 and M = e -τ n N , large large

(4.2) N (A(t C 0 ) ∩ B(0, N ), M ) ≥ t C 0 N M d .
This is the goal of the next paragraph.

4.2. Fourier decay and large coefficients. For a ∈ Z d and x ∈ T d , we denote by (a, x) → a, x ∈ T the natural pairing. Vectors in Z d = T d indexing Fourier coefficients are naturally understood as row vectors, so that for any g ∈ SL d (Z), we have a, gx = ag, x . Before we start the proof of ( large 4.2), we record an elementary lemma -not much more than the Cauchy-Schwarz inequality -which shows that the set of large Fourier coefficients of a measure has some additive structure. It will later be combined with the multiplicative properties of µ * n , allowing us to exploit the sum-product phenomenon for the study of the set of large Fourier coefficients. This approach to Fourier coefficients of multiplicative convolutions of measures goes back to the work of Bourgain and Konyagin bourgainkonyagin [START_REF] Bourgain | Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order[END_REF] on exponential sums in finite fields. We use the symbols and introduced in Section sc:sumprod

2.

lm:SpecHolder Lemma 4.3 (Additive structure of Fourier coefficients). Let µ be a Borel probability measure SL d (Z) and ν a Borel probability measure on

T d . If | µ * ν(a 0 )| ≥ t 0 > 0,
then for any integer k ≥ 1, the set

A = g ∈ M d (Z) | |ν(a 0 g)| ≥ t 2k 0 /2 satisfies µ k µ k (A) ≥ t 2k 0 2 . Proof. Observe that µ * ν(a 0 ) = T d Γ e( a 0 , gx ) dµ(g) dν(x) = T d Γ
e( a 0 g, x ) dµ(g) dν(x).

By Hölder's inequality,

t 2k 0 ≤ | µ * ν(a 0 )| 2k ≤ T d Γ e( a 0 g, x ) dµ(g) 2k dν(x) ≤ Γ 2k ν a 0 (g 1 + • • • + g k -g k+1 -• • • -g 2k ) dµ ⊗2k (g 1 , . . . , g 2k ) ≤ E |ν(a 0 g)| d µ k µ k (g) ≤ µ k µ k (A) + t 2k 0 2 µ k µ k (E \ A),
which finishes the proof of the lemma.

Combining the above observation and Theorem Proof of ( large 4.2). As before, for n ≥ 1, we let μn = (e -λ1n ) * µ n denote the rescaling of µ n = µ * n , and write D = dim E. Let α = 1/(4D + 2), and let s and c 0 be the constants given by Theorem thm:decay

3.19. Now fix δ = e -n s . Let C 1 = C 1 (D, α) from Lemma lm:dgammafull 4.4 below and set k = C 1 /c 0 so that ∀ξ ∈ E with δ -α ≤ ξ ≤ δ -2 , | μ k n μ k n (ξ)| ≤ ξ -C1 , This says that the measure μ k n μ k
n is regular at all scales between δ 2 and δ α . On the other hand, since | µ n * ν(a 0 )| ≥ t 0 , it follows from Lemma lm:SpecHolder

that the

set A = g ∈ E ∩ M d (Z) | |ν(a 0 g)| ≥ t 1 := t 2k 0 /2 satisfies µ k n µ k n (A) t 2k 0 .
Letting à = e -λ1n • A be the rescaling of A, we find

μ k n μ k n ( Ã) t 2k 0 .
From the large deviation estimate Theorem thm:LargeD

3.10

it:LargeDn (i), we also have

μ k n μ k n (E \ B E (0, δ -α )) µ e -c2n
for some

c 2 = c 2 (µ, s, k, α) > 0. Assuming n ≥ 4k c2 |log t 0 |, this implies μ k n μ k n ( Ã ∩ B E (0, δ -α )) t 2k 0 . So we can apply Lemma lm:dgammafull 4.4 to the restriction of μ k n μ k n to B E (0, δ -α ). Letting t 1 = t 2k 0 , we obtain x ∈ B E (0, δ -α ) such that N ( Ã ∩ B E (x, δ 1/2 ), δ) D t D+1 1 δ -D/2 .
Rescaling back, we find eq:AcapBx eq:AcapBx

(4.3) N A ∩ B E (e λ1n x, N 0 ), M 0 D t 2 N 0 M 0 D , where t 2 = t D+1 1 , N 0 = e (λ1-1 2s )n and M 0 = e -n 2s N 0 . Note that N = N 0 a 0 and M = M 0 a 0 .
Consider the map ϕ 0 :

E → R d , g → a 0 g. Letting A = A ∩ B E (e λ1n
x, N 0 ), we have eq:A'fiber eq:A'fiber

(4.4) N (A , M 0 ) ≤ N ϕ 0 (A ), M max b∈ϕ0(A ) N A ∩ ϕ -1 0 (B(b, M )), M 0 .
We claim that eq:A'capW eq:A'capW

(4.5) max b∈ϕ0(A ) N A ∩ ϕ -1 0 (B(b, M )), M 0 E N 0 M 0 D-d Evidently ϕ 0 E a 0 . Let W 0 = ker ϕ 0 . Since G acts irreducibly on R d , ϕ 0 is surjective and hence dim(W 0 ) = D -d. The restriction ϕ 0|W ⊥ 0 : W ⊥ 0 → R d is bijective. Moreover, by a compactness argument, ϕ -1 0|W ⊥ 0 E a 0 -1 .
Consequently, for any y ∈ E,

ϕ -1 0 (B(ϕ 0 (y), M )) ⊂ y + W (O E (M0)) 0 . Hence N A ∩ ϕ -1 0 (B(ϕ 0 (y), M )), M 0 ≤ N B E (0, N 0 ) ∩ W (O E (M0)) 0 , M 0 E N 0 M 0 D-d
, which proves the claim ( eq:A'capW 4.5). From ( eq:AcapBx

4.3), (

eq:A'fiber

4.4) and (

eq:A'capW 4.5), we get

N ϕ 0 (A ), M E t 2 N M d .
By definition of A , we have ϕ 0 (A ) ⊂ A(t 1 ) ∩ B(b, ϕ 0 N 0 ), where b = e λ1n a 0 x and ϕ 0 N 0 E N . This is almost what we want, except that the ball B(b, N ) is not centered at the origin. To recenter that ball, we make use once more of the additive properties of the set of large Fourier coefficients. Choosing the densest ball of radius N/2 inside B(b, N ), we get some b ∈ R d such that

N A(t 1 ) ∩ B(b , N 2 ), M E t 2 N M d .
Choose an M -separated subset A 1 ⊂ A(t 1 ) ∩ B(b , N/2) of cardinality Thus, there exists a 2 ∈ A 1 such that

|A 1 | N A(t 1 ) ∩ B(b , N 2
|A 1 | t 2 1 16 ≤ a1∈A1 |ν(a 1 -a 2 )|. Set A 2 = (A 1 -a 2 ) ∩ A t 2 1 32 , we have |A 2 | ≥ t 2 1 32 |A 1 | and A 2 ⊂ B(0, N ). It follows that N A t 2 1 32 ∩ B(0, N ), M E t 2 1 t 2 N M d .
This concludes our proof.

The next lemma is the regularity statement we need for measures on the Euclidean space that have a strong Fourier decay. It essentially states that if a set in R d carries a large proportion of a measure with small Fourier coefficients at all frequencies between δ -α and δ -1-α , then we can find a ball of radius δ O(α) in the set on which the measure is comparable to the Lebesgue measure at scale δ. Proof. Let ϕ : R D → R be a nonnegative smooth function supported on B(0, 1)

such that R D ϕ = 1. Set ϕ δ (x) = δ -D ϕ(δ -1 x), ∀x ∈ R D . Note that ∀ξ ∈ R D , ϕ δ (ξ) = φ(δξ).
Since ϕ is smooth, for any C 2 > 0, we have eq:hatphi eq:hatphi

(4.6) ∀ξ ∈ R D , | φ(ξ)| C2 (1 + ξ ) -C2 .
Define µ δ = µ ϕ δ , viewed either as a measure or as a smooth function on R D . Clearly, µ δ (A (δ) ) ≥ µ(A) ≥ t. Let c > 0 be a small constant depending on D and α to be determined later. Assume δ < ct and set ρ = cδ β t. Let (B i ) 1≤i≤imax be an essentially disjoint covering of B(0, δ -α ) by closed balls of radius ρ. In other words, the intersection multiplicity of the covering is at most C d = O d (1), so that in particular the number of balls is at most i max = O d (δ Dα ρ -D ). Consider

I = 1 ≤ i ≤ i max | µ δ (A (δ) ∩ B i ) µ δ (B i ) ≥ t 2C d .
Using the finite multiplicity of the covering, we infer that i∈I µ δ (B i ) ≥ t/2. Hence there exists i ∈ I such that

µ δ (B i ) t i max d tδ Dα ρ -D .
We fix this i from now on. Define We have µ δ (B i ) ≤ M |B i | and hence eq:MforMax eq:MforMax

µ δ (x) = µ ϕ δ (x) = R D e(-ξ, x )μ(ξ) ϕ δ (ξ) dξ
Thus,

|µ δ (x) -µ δ (x 0 )| ≤ R D |1 -e( ξ, x -x 0 )||μ(ξ)|| ϕ δ (ξ)| dξ ≤ T 1 + 2T 2 + 2T 3 ,
where

T 1 = ξ ≤δ -α |1 -e( ξ, x -x 0 )| dξ d ξ ≤δ -α ξ x -x 0 dξ d δ -(D+1)α ρ, T 2 = δ -α ≤ ξ ≤δ -1-α |μ(ξ)| dξ d δ -α ≤ ξ ≤δ -1-α ξ -C1 dξ d δ (C1-D)α C 1 -D , T 3 = ξ ≥δ -1-α | φ(δξ)| dξ D,C2 ξ ≥δ -1-α δ -C2 ξ -C2 dξ D,C2 δ (C2-D)α-D C 2 -D .
In the last line, we used ( eq:hatphi This implies M/m ≤ 2 provided that c is chosen small enough according to D and α. Remembering i ∈ I, we have

t d µ δ (A (δ) ∩ B i ) µ δ (B i ) ≤ M m |A (δ) ∩ B i | |B i | .
Hence

N (A ∩ B i , δ) d δ -D |A (δ) ∩ B i | d tδ -D |B i | d tδ -D ρ D = c D t D+1 δ Dβ-D .
Let x ∈ R D be the center of B i . Then A ∩ B i ⊂ A ∩ B(x, δ β ) and hence

N (A ∩ B(x, δ β ), δ) D,α t D+1 δ Dβ-D .

Concentration near rational points sc:rational

In this section, we finish the proof of Theorem thm:main 1.2 from the introduction. The argument follows closely the one given in Section 7 of BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], but some modifications are required since we cannot make use of the proximality assumption.

In all this section, unless stated otherwise, µ denotes a probability measure on SL d (Z), d ≥ 2. The subsemigroup generated by µ is denoted by Γ, its Zariski closure by G, and we write G = G(R) for the set of real points. We assume that: We also let E be the subalgebra of M d (R) generated by G. Finally, for n ∈ N, we write µ n = µ * n for the law of the random walk at time n.

We shall divide the proof of Theorem thm:main 1.2 into three parts. First, one observes that given a probability measure ν on T d , the sequence of measures µ n * ν satisfies a diophantine property: if it gives much weight to a ball of small radius, then the ball must contain a rational point with small denominator. Second, starting the separated set X around which, by Proposition pr:step1 4.1, µ n * ν is concentrated, one goes backwards along the random walk in order to increase the concentration of the measure around the set X, until one can apply the diophantine property to conclude that µ n-m * ν is concentrated near some rational points with bounded denominator. The last part, concluding the proof, is again going backwards along the random walk, to show that if µ n * ν concentrates near the set of rational points of bounded height, then ν is even more concentrated near that set.

5.1.

An almost diophantine property. The key to obtain the concentration near rational points is the following almost diophantine property of the sequence of measures µ n * ν, n ∈ N. Given Q ≥ 1 and ρ > 0, we denote by W Q the set of rational points in T d with denominator at most Q, and by W (ρ) Q its ρ-neighborhood. pr:dioph Proposition 5.1 (Almost diophantine property). Let µ be a probability measure on SL d (Z), d ≥ 2, with some finite exponential moment. Assume that µ acts strongly irreducibly on R d .

There exist constants C ≥ 0 and η > 0 depending only on µ, such that for every probability measure ν on T d , for every x ∈ T d , every ρ > 0, and every n ≥ C|log ρ|,

(µ n * ν)(B(x, ρ)) ≥ ρ η =⇒ x ∈ W (ρ 9/10 ) ρ -1/10 .
Proof. By Lemma lm:notaffine 3.14, the manifold G × G is not included in any proper affine subspace of E × E and therefore the set

{g 1 + • • • + g d -h 1 -• • • -h d | g i , h i ∈ G}
contains a non-empty open set in E. This implies in particular that the map

(g i , h i ) → det(g 1 + • • • + g d -h 1 -• • • -h d ) is not identically zero on G d × G d .

By Proposition

pr:gapEscape 3.8, we infer that there exists c > 0 such that for every m large enough,

µ ⊗2d m ({(g i , h i ) 1≤i≤d | det( g i - h i ) = 0}) ≤ e -cm .
and n -m ≥ Cm,

(µ n-m * ν) W (Q -8 ) Q ≥ t C 0 , for some Q ∈ [e m C , e Cm ].
The concentration statement given by Proposition pr:step1 4.1 is not strong enough for a direct application of the diophantine property. We first need Lemma and the proof is also the same, with some minor modifications to avoid the use of the proximality assumption; we include it nonetheless, for readability. lm:rewind Lemma 5.3. Given ε > 0, there exist c > 0 and m 0 ∈ N so that for m ≥ m 0 , the following holds for every probability measure ν on T d . Given scales r, ρ > 0 such that e dλ1m ρ < r, there are scales r 1 = e -m(λ1+ε) and ρ 1 = e -m(λ1-ε) ρ, so that for every r-separated set X ⊂ T d , one can construct an r 1 -separated set

X 1 ⊂ T d with ν(X (ρ1) 1 ) ≥ (µ m * ν)(X (ρ) ) d -e -cm .
Proof. First write

(µ m * ν(X (ρ) )) d = ( g∈Γ µ m (g)ν(g -1 X (ρ) )) d = ( g∈Γ µ m (g)1 g -1 X (ρ) (x) dν(x)) d ≤ ( g∈Γ µ m (g)1 g -1 X (ρ) (x)) d dν(x) = g1,...,g d ∈Γ µ m (g 1 ) . . . µ m (g d )ν(g -1 1 X (ρ) ∩ • • • ∩ g -1 d X (ρ) ).
This implies that the set of d-tuples (g i ) 1≤i≤d such that inter inter

(5.2) ν(g -1 1 X (ρ) ∩ • • • ∩ g -1 d X (ρ) ) ≥ (µ m * ν)(X (ρ) ) d -e -
(5.3) ∀i = 1, . . . , d, g -1 i ≤ e (-λ d +ε)m and placegi placegi (5.4) ∀v ∈ R d \ {0}, max i g i v v ≥ e (λ1-ε)m .
We fix such elements g 1 , . . . , g d for the rest of the proof. Without loss of generality, we may assume that ε > 0 is so small that

λ 1 -λ d + 3ε < dλ 1 .
We claim then that the set g

-1 1 X (ρ) ∩ • • • ∩ g -1 d X (ρ)
is included in a union of at most |X| balls of radius ρ 1 = e -m(λ1-ε) ρ: Indeed, from ( normgi 5.3) one finds -drawing a picture of X (ρ) and g -1 i X (ρ) -that given x ∈ X and i ≥ 1, the set g -1 1 B(x, ρ) meets at most one component g -1 i B(y, ρ), y ∈ X. Therefore, there are at most |X| non-empty intersections g -1 1 B(x 1 , ρ) ∩ . . . g -1 d B(x d , ρ), for x 1 , . . . , x d ∈ X. Finally, if x, y lie inside such an intersection, then, for each i, g i (x -y) ≤ ρ, and ( placegi 5.4) implies that x -y ≤ e -m(λ1-ε) ρ = ρ 1 . Thus, each intersection g -1 1 B(x 1 , ρ) ∩ . . . g -1 d B(x d , ρ) is included in a ball of radius ρ 1 , and the proposition follows.

We now state and prove the large deviation estimate use in the above argument. lm:place Lemma 5.4. Let µ be a probability measure on SL d (R) with some finite exponential moment, and assume that the semigroup Γ generated by µ acts strongly irreducibly on R d . Then, for every ε > 0, there exists c > 0 such that for every large enough m ∈ N,

µ ⊗d m (g 1 , . . . , g d ) | ∀v ∈ R d \ {0}, max i g i v v ≥ e (λ1-ε)m ≥ 1 -e -cm .
Remark 4. If one assumes that µ is supported on SL d (Z), and replaces d by d 2 , then this lemma follows directly from Proposition pr:NCdet 3.2. This particular case would be sufficient for our purposes.

Proof. In this proof, c denotes a small positive constant, depending on ε, and whose value may vary from one line to the other. Let r denote the proximality dimension of G. For g ∈ G, we shall consider its Cartan decomposition g = k diag(σ 1 (g), . . (i) if g 1 , . . . , g d are independent random variables with law µ m , then with probability at least 1 -e -cm , (5.6) ∀i ∈ {1, . . . , d}, g i ≥ e m(λ1-ε) .

For a subspace W ≤ R d , we let Nbd(W, ρ) denote the ρ-neighborhood of W in R d . It follows from the above that the lemma will be proved -with 4dε instead of ε -if we can show that with probability at least 1 -e -cm , the intersection

d i=1 Nbd(W - gi , e -4dεm )
reduces to a ball of radius 1 2 . For that, we construct inductively for k = 1, . . . , d -r + 1 a linear subspace W k of dimension d -r + 1 -k, depending on g 1 , . . . , g k , such that k i=1 Nbd(W - gi , e -4dεm ) ⊂ Nbd(W k , e -4(d+1-k)εm ).

At each step W k+1 is constructed in terms of W k and g k+1 and the construction is possible with probability 1 -e -cm . For k = 1, one may simply take W 1 = W - g1 . Then, suppose W k has been constructed, and let Rw ⊂ W k be any line. By Theorem Since by a theorem of Guivarc'h and Raugi GuivarchRaugi [START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF], λ r > λ r+1 , we deduce from the above that, provided ε > 0 is small enough, d(Rw, W - g k+1 ) ≥ e -3εm . This implies that there exists a proper subspace W k+1 < W k such that Nbd(W k , e -4(d-k+1)εm ) ∩ Nbd(W - g k+1 , e -4dεm ) ⊂ Nbd(W k+1 , e -4(d-k)εm ). This proves what we want.

To prove Proposition pr:step2 5.2, we proceed as follows. Applying first Proposition pr:step1

4.1, we shall obtain m 0 ∈ N and scales ρ 0 and r 0 , together with an r 0 -separated set

X 0 ⊂ T d such that µ n-m0 * ν(X (ρ0) 0 ) ≥ t C 0 .
The idea is then to reduce the radius of the balls, by an iterated application of Lemma lm:rewind 5.3. We shall thus obtain an increasing sequence of integers m k and a decreasing sequence of scales ρ k and r k , k = 1, 2, . . . together with an r k -separated set

X k ⊂ T d such that |X k | ≤ |X 0 | and µ n-m k * ν(X (ρ k ) k ) ≥ t C k 0 .
Once we arrive at a scale ρ k such that

|X k | ≤ |X 0 | ≤ ρ -η 2 k ,
we shall be able to use the diophantine property of the random walk to conclude. Now let us turn to the detailed proof.

Proof of Proposition

≥ C 0 | log t 0 |, τ m 0 4dλ 1 ≤ m + ≤ τ m 0 2dλ 1 and k = 8d 2 λ 1 ητ = O µ (1).
This is feasible provided m ≥ C|log t 0 |, where C ≥ 0 depends on µ via the constants C 0 , τ , etc. Note that within constants depending only on µ, m m 0 m + .

By Proposition

pr:step1

4.1 applied to

µ n * ν = µ m0 * (µ n-m0 * ν),
there exist scales ρ 0 = e -m0(λ1-1 2s ) and r 0 = e τ m0 ρ 0 together with an r 0 -separated subset X 0 ⊂ T d such that

(µ n-m0 * ν)(X (ρ0) 0 ) ≥ t C0 0 . Choose ε > 0 such that 2kε < dλ 1 so that kεm + < τ m 0 2 and apply Lemma lm:rewind 5.3 to µ n-m0 * ν = µ m+ * (µ n-m0-m+ * ν).
This is allowed since by our choice of parameters e dλ1m+ ρ 0 ≤ e τ m 0 2 ρ 0 < r 0 . This yields scales ρ 1 = e -m+(λ1-ε) ρ 0 and r 1 = e -m+(λ1+ε) r 0 together with an r 1 -separated subset

X 1 such that |X 1 | ≤ |X 0 | and (µ n-m0-m+ * ν)(X (ρ1) 1 ) ≥ t C1
0 , provided m is large enough to ensure that e -cm+ < t C1 0 . We may repeat this procedure at least k times, and therefore obtain a sequence of scales defined inductively by ρ i+1 = e -m+(λ1-ε) ρ i and r i+1 = e -m+(λ1+ε) r i .

Indeed, our choice of ε ensures that for every i ≤ k, ρ i ≤ e -τ m 0 2 r i ≤ e dλ1m+ r i .

In the end, we obtain scales ρ k and r k , and a set X k with

|X k | ≤ |X 0 | ≤ e dλ1m0 such that xklarge xklarge (5.8) (µ n-m * ν)(X (ρ k ) k ) ≥ t C k 0 . Moreover, ρ k = e -km+(λ1-ε) ρ 0 ≤ e -km+ λ 1 2 ≤ e -2dλ 1 m 0 η so that |X k | ≤ ρ -η 2
k . Therefore, adjusting slightly the values of the constants, we may restrict X k to the points satisfying (µ n-m * ν)(B(x, ρ k )) ≥ ρ η k , while preserving (

(µ m * ν)(W (e -(λ 1 -ε)m ρ) Q ) ≥ ν(W (ρ) Q ) -e -ωm .
The proof of this proposition is based on the following lemma. lm:nombre Lemma 5.6. Let µ be a probability measure on SL d (R) with some finite exponential moment and acting strongly irreducibly on R d . Given ε > 0, there exists θ > 0 such that the following holds for every integer m sufficiently large.

Let A be a subset of SL d (R) such that µ m (A) ≥ e -θm . There exists a subset G = {g i } 1≤i≤k of cardinality k ≥ e θm in A such that for every subset {g i1 , . . . , g

i d } of d elements of G, for every v in R d , max 1≤j≤d g ij v ≥ e m(λ1-ε) v . Proof. Having fixed ε > 0, let T = {(h 1 , . . . , h d ) | ∀v ∈ R d \ {0}, max i h i v v < e (λ1-ε)m }.
By Lemma lm:place 5.4, there exists c > 0 such that for every large enough m, µ ⊗d m (T ) ≤ e -cm . We shall prove that the lemma holds with θ = c d2 d+1 . Let A 1 = {g ∈ A | µ ⊗d-1 m ({(h 2 , . . . , h d ) | (g, h 2 , . . . , h d ) ∈ T }) ≥ e -cm/2 }.

Then

e -cm ≥ µ ⊗d m (T ) ≥ e -cm/2 µ m (A 1 ), and therefore µ m (A 1 ) ≤ e -cm/2 . To construct G, we first choose g 1 ∈ A \ A 1 ; this is possible because θ < c/2. Let ) ≤ e -cm/4 . We may therefore pick an element g 2 ∈ A such that g 2 ∈ A 1 ∪ A 2 (g 1 ). Then set A 3 (g 1 , g 2 ) = {g | µ ⊗d-3 m ({(h 4 , . . . , h d ) | (g 1 , g 2 , g, h 4 , . . . , h d ) ∈ T }) ≥ e -cm/8 } for which it is readily checked, using the fact g 2 ∈ A 1 (g 1 ), that µ m (A 3 (g 1 , g 2 )) ≤ e -cm/8 . This allows us to pick g 3 ∈ A such that g 3 ∈ A 1 ∪ A 2 (g 1 ) ∪ A 2 (g 2 ) ∪ A 3 (g 1 , g 2 ). Following this procedure, the elements g 1 , g 2 , g 3 , . . . of G are constructed inductively. Once g 1 , . . . , g k have been chosen, one picks g k+1 ∈ A outside the union of all subsets A r (g i1 , . . . , g ir-1 ) = {g | µ ⊗d-r m ({(h r+1 , . . . , h d ) | (g i1 , . . . , g ir-1 , g, h r+1 , . . . , g d ) ∈ T }) ≥ e -cm/2 r }, where (g i1 , . . . , g ir-1 ) can be any subset of (g 1 , . . . , g k ) with at most d elements. By convention, for r = d, write

A d (g i1 , . . . , g i d-1 ) = {g | (g i1 , . . . , g i d-1 , g) ∈ T }.

Just as above, one checks by induction, using g ir-1 ∈ A r-1 (g i1 , . . . , g ir-2 ), that µ m (A r (g i1 , . . . , g ir-1 )) ≤ e -cm/2 r ≤ e -cm/2 d .

Thus, at step k, the union of all subsets A r (g i1 , . . . , g ir ) to be avoided has measure at most

[1 + k 1 + • • • + k d ]e -cm/2 d ≤ k d e -cm/2 d .
So the procedure can go on as long as k d e -cm/2 d < µ m (A). Since µ m (A) ≥ e -θm = e -cm/2 d+1 , one can at least reach some k ≥ e cm d2 d+1 , which proves the lemma.

The rest of the proof of Proposition For such elements g 1 , . . . , g d , i.e. u = g -1 i (x i + v i ) + O(e dλ1m ρ). But the points g -1 i (x i + v i ) are rational with denominator at most Q, so that they are at least Q -2 away from one another. Since e dλ1m ρ < Q -2 , this shows that there exists u 0 ∈ R d , rational with denominator at most Q such that for each i, g i u 0 = x i + v i . Coming back to ( ugi 5.9) above, we find ) -e -cn ≥ t C -e -cnj e -ωmj .

g -1 1 W (ρ) Q ∩ . . . g -1 d W (ρ) Q = x1,...,x d ∈W Q g -1 1 B(x 1 , ρ) ∩ • • • ∩ g -1 d B(x d , ρ).
To conclude, it suffices to show that j e -ωmj is bounded above by t C /2 provided m 0 (or Q) is chosen large enough. Now the recurrence relation defining the numbers m i shows that

ρ i+1 = e -λmi+1 ρ i et e dλ1mi+1 ρ i Q -2 .
This implies

e dλ1mi+1 Q -2 ρ -1 i = Q -2 e λmi ρ -1 i-1 = Q -2 e λ(mi+•••+m1) ρ -1 0 e λmi e dλ1mi
and therefore

m i+1 = dλ 1 + λ dλ 1 m i + O(1).
Since dλ1+λ dλ1 > 1, this shows that the sum e -ωmi converges, which is what we wanted to show.

Conclusion sc:conclusion

To conclude this paper, we mention one application of our main theorem, and then give some possible further directions of research, some of which we hope to address in publications to come. ss:exp 6.1. Expansion in simple groups modulo arbitrary integers. In Section sc:noncon 3, we made use of the result of Salehi Golsefidy and Varjú SGV [START_REF] Golsefidy | Expansion in perfect groups[END_REF] about expansion in semisimple groups modulo prime -or square-free -numbers. In a reverse direction, it was observed by Bourgain and Varjú bv [START_REF] Bourgain | Expansion in SL d (Z/qZ), q arbitrary[END_REF] that the quantitative equidistibution of linear random walks on the torus of Bourgain, Furman, Lindenstrauss and Mozes BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] could be used to derive some expansion results in SL d (Z/qZ), where q runs over all natural integers. Because of the proximality assumption required by BFLM [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], their argument could only apply to R-split simple Q-groups, such as SL d . With Theorem thm:main 1.2 at hand, we can now generalize their result to any simple Q-group. Theorem 6.1 (Expansion in simple groups modulo arbitrary integers). Let S be a finite subset of GL d (Z), and Γ the subgroup generated by S. If the Zariski closure of Γ is a connected simple algebraic group, then the family of Cayley graphs G(π q (Γ), π q (S)) q∈N is a family of expanders .

As observed by SGV [START_REF] Golsefidy | Expansion in perfect groups[END_REF], one should expect the theorem to hold with the weaker assumption that the Zariski closure of Γ is perfect. To prove such a result, if one wants to exploit some equidistribution result on the torus similar to Theorem thm:main 1.2, one should relax the irreducibility assumption, which leads us to the second point of this conclusion. ] that in order to prove equidistribution of a linear random walk on the torus, it was more natural only to assume the action to be irreducible on Q d , as opposed to R d . Indeed, for example, if the group generated by the random walk is semisimple, this implies that every proper closed invariant subset is a finite set of rational points. Similarly, Theorem thm:main 1.2 should remain valid if one only assumes the irreducibility of the action of Γ on Q d , as long as the Zariski closure of Γ is semisimple.

The general approach used here should work in this setting, but there is one important difference: the algebra E generated by Γ will no longer be simple, but only semisimple. In particular, the rescaled measure μn studied in Section sc:noncon 3 may very well be concentrated on a proper ideal of E. One therefore needs to modify several of our arguments to adapt the proof to this more general setting.

We hope to address this issue in forthcoming work in collaboration with Elon Lindenstrauss.

6.3. The two other assumptions. First, we believe that the Theorem thm:main 1.2 is still valid even if one does not require the group G to be Zariski connected. In fact, many arguments in our proof still works without this assumption, but, as is the case without the irreducibility assumption, the rescaled measures μn may concentrate near a proper subspace of E: the algebra generated by the connected component of G. This leads to several technical difficulties when trying to prove a flattening statement.

Second, it would be an interesting problem to determine what moment conditions are really necessary in order to have the convergence statement of Theorems

Theorem 1 . 1 (

 11 Equidistribution on the torus). Let d ≥ 2. Let µ be a probability measure on SL d (Z). Denote by Γ the subsemigroup generated by µ, and by G < SL d the Zariski closure of Γ. Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The only subspaces of R d preserved by Γ are {0} and R d ; it:SA3 (c) The algebraic group G is Zariski connected.

[ 7 ,

 7 Proof of Theorem C] sometimes known as the additive-multiplicative Balog-Szemerédi-Gowers theorem.

  On the other hand, writing c = (a , b ), we have a A ≈ A b . Hence a A b -1 b ≈ A b and then a A b -1 b ≈ aA and finally eq:countersumprod eq:countersumprod

  are immediate.To check the remaining assumptions, write, for b ∈ E,B 1 (b) = {a ∈ E | (a, b) ∈ B}.By assumption it:spB (vi) of the proposition we are trying to prove, we can pick b

Bourgain2010

[ 8 ,

 8 Theorem 7]. The proof is essentially the same. A detailed proof is implicitly contained in He_these [24, Lemma 2.11]. For a finite Borel measure ν on E and an integer ≥ 1, we denote by ν = ν • • • ν times the -fold additive convolution of ν.

Lemma 2 . 7 .

 27 Let E be a finite-dimensional real algebra. Let ≥ 1 be an integer, ν a Borel probability measure on E, and set µ = ν ν .

Proposition 3 . 1 (

 31 Non-concentration on affine subspaces). Let µ be a probability measure on SL d (Z), for some d ≥ 2. Denote by Γ the subsemigroup generated by µ, and by G the Zariski closure of Γ in SL d . Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c) The algebraic group G is Zariski connected. There exists κ > 0 such that for every proper affine subspace

Proposition 3 . 2 (

 32 Non-concentration on subvarieties). Let µ be a probability measure on SL d (Z), for some d ≥ 2. Let Γ denote the subsemigroup generated by µ, G the Zariski closure of Γ in SL d , and λ 1 the top Lyapunov exponent of µ. Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c) The algebraic group G is Zariski connected.

SGV [ 21 ,k µ 2 0

 212 Theorem 1]. thm:specgap Theorem 3.3 (Spectral gap theorem). Let d ≥ 2 and let µ be a probability measure on SL d (Z) such that the Zariski closed subgroup G generated by µ is semisimple. Then there exists a constant c > 0 and an integer k such that for every prime number p, T (Gp(Fp)) ≤ 1 -c. Remark 3. Another way to state the above theorem is to say that the spectral radius of the operator T µ restricted to 2 0 (G p (F p )) is bounded above by 1 -c, for every prime number p.For completeness, we now explain how to derive the above theorem from SGV [21,Theorem 1]. The argument uses the following two lemmata.

Lemma 3 . 4 .

 34 Let Γ be a subsemigroup of SL d (R) whose Zariski closure G is semisimple. There exists a finite subset S ⊂ Γ such that the semigroup generated by S is Zariski dense in G.

Borel [ 5 ,

 5 Theorem 6.8 and Corollary 14.11], the quotient G/H is a semisimple linear algebraic group such that the projection π : G → G/H is a morphism of algebraic groups.

nori [ 30 ,

 30 Theorem 5.2], the group Γ generated by S is dense in some open subgroup Ω of G(Q p ), for some prime number p. In other words, the union k≥1 S k is dense in Ω. Since the Lie algebra g of Ω satisfies [g, g] = g, the derived subgroup [Ω, Ω] contains an open subgroup Ω ; then, the increasing sequence of subsets (S k S -k ) k≥1 gets arbitrarily dense in Ω .

thm:specgap 3 . 3 .

 33 Let μ denote the image measure of µ by the map g → g -1 . By lemmata lm:Sfinite 3.4 and lm:S-kSk 3.5 above, there is k ≥ 1 such that the support of μ * k * µ * k contains a finite symmetric subset S which generates a Zariski dense subgroup in G. By SGV [21, Theorem 1] applied to S, there is c > 0 such that for any prime number p sufficiently large,

Lemma 3 . 7 (

 37 Strong irreducibility and semisimplicity). Let G be a Zariski closed subgroup of SL d (R) generated by elements of SL d (Z) and acting strongly irreducibly on R d . Then G is semisimple.

  pr:gapEscape Proposition 3.8. Let µ be a Borel probability measure on SL d (Z) such that the Zariski closed subgroup G generated by Supp µ is semisimple and connected.

  be the ideal of all polynomials with coefficients in Q and vanishing on V . Let I Z = I ∩ Z[X]. For any prime number p, let V p be the variety over F p defined by the ideal π p (I Z ) ⊂ F p [X]. By the Bertini-Noether theorem FriedMoshe [19, Proposition 10.4.2],

For

  g ∈ G, write its Cartan decomposition g = k diag(σ 1 (g), . . . , σ d (g)) , where k and are orthogonal matrices and σ 1 (g) ≥ • • • ≥ σ d (g) are the singular values of g. Define also V + g = k Span(e 1 , . . . , e r ) and W - g = -1 Span(e r+1 , . . . , e d ) where (e 1 , . . . , e d ) is the standard basis of R d . We first prove the proposition in the case where G is proximal, i.e. r = 1. Under this condition, we have BFLM [9, Lemma 4.1(2)] :

3. 4 .

 4 Escaping a small neighborhood of a subvariety. For an integer D ≥ 0, and a regular function f ∈ R[G] on G, we say that f has degree at most D if it can be represented by a polynomial on M d of degree at most D. Denote by R[G] ≤D the finite-dimensional subspace consisting of regular functions of degree at most D. We fix a norm on R[G] ≤D . lm:mundef<e Lemma 3.12. Let µ be a Borel probability measure on SL d (Z) having a finite exponential moment. Assume that the Zariski closed subgroup G generated by Supp µ is semisimple and connected.

Lemma 3 . 13 .

 313 Let µ be a Borel probability measure on SL d (Z) having a finite exponential moment. Assume that the Zariski closed subgroup G generated by Supp µ is semisimple and connected.

  lm:notaffine Lemma 3.14. Let G be a subgroup of GL d (R) acting irreducibly on R d , and E the associative subalgebra generated by G in M d (R). Then G is not contained in any proper affine subspace of E.

Lemma 3 . 15 .

 315 If a nonzero subalgebra of M d (R) does not preserve any proper nontrivial subspaces of R d , then it contains the multiplicative identity of M d (R).

VanderWaerden [ 35 ,

 35 Chapter XVI, §116]. By VanderWaerden [35, Chapter XVI, §117] 2 , W has a multiplicative identity 1

  a submodule of the the semisimple G-module M d (R) * , which is isomorphic to the sum of d copies of the simple G-module R d . It follows that E * is isomorphic to the sum of dim(E) d copies of R d . For each i = 1, . . . , dim(E) d , let π i : R ⊕ E * → R d denote the projection to the i-th R d -factor. Remembering

Proposition 3 . 16 .

 316 Let µ be a probability measure on SL d (Z), for some d ≥ 2. Let Γ denote the subsemigroup generated by µ, G the Zariski closure of Γ in SL d , and λ 1 the top Lyapunov exponent of µ. Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c) The algebraic group G is Zariski connected, and let the notation be as above.

3. 7 .

 7 Criterion to have nonzero component in modules of maximal Lyapunov exponent. In order to use Proposition pr:LDPpolynome

  Dnχ σ (a) ≤ nχ Mj (b) + O(1) and necessarily Dχ σ (a) ≤ χ Mj (b).

Theorem 3 . 19 (

 319 Fourier decay for μn ). Let µ be a probability measure on SL d (Z), d ≥ 2. Let Γ denote the subsemigroup generated by µ, G the Zariski closure of Γ in SL d , and λ 1 the top Lyapunov exponent of µ. Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c) The algebraic group G is Zariski connected. For every α > 0, there exists s ≥ 1 and c 0 > 0 such that for every n ≥ n 0 , ∀ξ ∈ E * with e αn s ≤ ξ ≤ e n sα , | μn (ξ)| ≤ e -c0n .

n

  and η = η μ D n . By Theorem thm:LargeD

e -αn 2 ξ 4 .Proposition 4 . 1 (

 2441 ≤ ξx ≤ e αn 2 ξ , because μr has bounded exponential moment. The set of large Fourier coefficients sc:fourier Starting from Theorem thm:decay 3.19, we now use some Fourier analysis to show a first intermediate statement towards Theorem First step: concentration and separation). Let µ be a probability measure on SL d (Z), d ≥ 2. Denote by Γ the subsemigroup generated by µ, and by G the Zariski closure of Γ in SL d . Assume that: it:SA1 (a) The measure µ has a finite exponential moment; it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c) The algebraic group G is Zariski connected.

|ν(a 1 -

 1 ), M and such that all Fourier coefficients ν(a), for a ∈ A 1 , fall into the same quadrant of C. Then |A 1 a 2 )|

Lemma 4 . 4 (

 44 Regularity from Fourier decay). Given D ≥ 1 and α > 0, there exist constants c = c(D, α) > 0 and C 1 = C 1 (D, α) > 0 such that the following holds for all 0 < δ < ct. Let µ be a Borel measure on R D , of total mass µ(R D ) ≤ 1. Let A be a subset of R D . Assume (i) Supp(µ) ⊂ B(0, δ -α ),(ii) ∀ξ ∈ R D , with δ -α ≤ ξ ≤ δ -1-α , |μ(ξ)| ≤ ξ -C1 , (iii) µ(A) ≥ t. Then there exists x ∈ R D such that N (A ∩ B(x, δ β ), δ) ≥ ct D+1 δ β δ D ,where β = (2D + 1)α.

M

  = max x∈Bi µ δ (x) and m = min x∈Bi µ δ (x).

(4. 7 )

 7 M d tδ Dα . Let x 0 ∈ B i such that µ δ (x 0 ) = M . By the Plancherel theorem, for any x ∈ B i ,

4. 6

 6 ). Picking β = (2D + 1)α, C 1 = Dα+1 α + D and C 2 = Dα+D+1 α + D and putting these inequalities together, we obtain, remembering ( eq:MforMax 4.7) and δ < ct, M -m D,α ctδ Dα + δ Dα+1 D,α cM.

  The measure µ has a finite exponential moment;it:SA2 (b) The action of Γ on R d is irreducible; it:SA3 (c)The algebraic group G is Zariski connected.

[ 9 ,

 9 to bootstrap concentration. It is exactly the same statement as BFLM Proposition 7.2],

  . , σ d (g)) , where k and are orthogonal matrices and σ 1 (g) ≥ • • • ≥ σ d (g) are the singular values of g. Define also W - g = -1 Span(e r+1 , . . . , e d ) where (e 1 , . . . , e d ) is the standard basis of R d . From BFLM [9, Lemma 4.1(2)], for every non-zero v ∈ R d , eq:sigmaV-eq:sigmaV-

  (iii), with probability at least 1 -e -cm g k+1 w w ≥ e (λ1-ε)m . , with probability 1 -e -cm , eq:sigmalambda eq:sigmalambda (5.7) ∀j ∈ {1, . . . , d}, | 1 m log σ j (g k+1 ) -λ j | ≤ ε, and by BFLM [9, Lemma 4.1(2)], gw w ≤ g d(Rw, W - g k+1 ) + σ r+1 (g k+1 ).

pr:step2 5 . 2 .

 52 Let C 0 and τ > 0 be the constants given by Proposition pr:step1 4.1. and C and η > 0 the ones given by Proposition pr:dioph 5.1. Then write m = m 0 + km + , where m 0

xklarge 5 . 8 ). 5 . 3 .[ 9 ,[ 9 ,

 585399 . If C was chosen large enough, then n-m ≥ Cm ≥ C |log ρ k |, and we may conclude by Proposition End of the proof of Proposition pr:general 5.7: near rational points. The end of the proof of Proposition pr:general 5.7 is based on an argument similar in spirit to the one used in Lemma lm:rewind 5.3, to bootstrap concentration. The proposition we shall need is again taken from BFLM[START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], where it appears as BFLMProposition 7.4]. The proof we present follows closely the one given in BFLM[START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF], but the key Lemma lm:nombre 5.6 below, analogous to BFLM Lemma 7.10], is proved using a new argument, which avoids using a regularity property of the µ-stationary measure on the projective space, only available with a proximality assumption. pr:rewindbis Proposition 5.5. Let µ be a probability measure on SL d (Z) with some finite exponential moment and acting strongly irreducibly on R d . Given ε > 0, there exist m * and ω > 0 such that if ρ > 0, Q ≥ 1 and m ≥ m * satisfy e dλ1m ρ < Q -2 , and ν is any probability measure on T d , then

A 2 (

 2 g 1 ) = {g ∈ A | µ ⊗d-2 m ({(h 3 , . . . , h d ) | (g 1 , g, h 3 . . . , h d ) ∈ T }) ≥ e -cm/4 }. Since g 1 ∈ A 1 , we have e -cm/2 ≥ µ ⊗d-1 m ({(h 2 , . . . , h d ) | (g 1 , h 2 , . . . , h d ) ∈ T }) ≥ e -cm/4 µ m (A 2 (g 1 )), whence µ m (A 2 (g 1 )

5 .A

 5 Once more, write(µ m * ν)(W = {g | ν(g -1 W (ρ) Q ) ≥ (µ m * ν)(B) -e -θm } satisfies µ m (A) ≥ e -θm .Using the large deviation estimate for g -1 , we may reduce A without any significant loss of µ m -measure so that for every g in A,g -1 ≤ e m(λ d +ε) ≤ 1 2 e mdλ1 .By Lemma lm:nombre 5.6, there exists a subset G ⊂ A of cardinality at least e θm such that for any distinct elements g 1 , . . . , g d in G, for every v ∈ R d , max 1≤i≤d g i v ≥ e m(λ1-ε) v .

  Now, if u ∈ R d represents an element of g -1 1 B(x 1 , ρ) ∩ • • • ∩ g -1 d B(x d , ρ), then, for some vectors v i ∈ Z d , ugi ugi(5.9)g i u = x i + v i + O(ρ), i = 1, . . . , d

  g i (u -u 0 ) ≤ ρ, i = 1, . . . , dand by definition of the subset G,u -u 0 ≤ e -m(λ1-ε) ρ. m(λ 1 -ε) ρ) Q ; g ∈ G} has intersection multiplicity less than d. Therefore, g∈G µ m (g -1 W (ρ) Q \ W (e -m(λ 1 -ε) ρ) Q ) ≤ d,and as card G ≥ e θm , there must exist g in G such thatµ m (g -1 W (ρ) Q \ W (e -m(λ 1 -ε) ρ) Q ) ≤ de -θm .and therefore, for every ε > 0, for n large enough ν(W (e -(λ-2ε)n Q)) ≥ ν r (W (e -(λ-ε)n ) Q

6. 2 .

 2 Without irreducibility. It was observed by Benoist and Quint bq2 [3, Corollary 1.

4

 4

2 . 6 . 4 .

 264 It seems plausible for example that Theorem thm:easy 1.1 holds with the weaker assumption of a moment of order 1: log g dµ(g) < ∞. Even a counter-example to Theorem thm:easy 1.1 without any moment condition would be interesting. Spaces of lattices. Given the results of Benoist and Quint bq2 [3] classifying stationary measures on the space of lattices SL d (R)/ SL d (Z), it is very natural to ask whether one can obtain an analog of Theorem thm:main 1.2 in this setting. Even the following qualitative equidistribution problem is still open bqintro [2, §5.4. Question 3]. Let µ be a measure on SL d (R) generating a Zariski dense subgroup Γ, and x a point in SL d (R)/ SL d (Z) with infinite Γ-orbit. Show that the sequence of measures (µ n * δ x ) n≥1 converges to the Haar measure as n goes to infinity.

Thus item

	follows from assumption Finally, assume for contradiction that item it:spmu3 (v) of Proposition pr:sumprod 2.3. it:spPhi3	it:spPhi2 (v)

  we see that the result follows from (

	eq:normgv 3.3). BougerolLacroix [6, Proof of Theorem V.6.2] to reduce the proposition to the proximal case. Namely, by We now use Bougerol's trick Breuillard [15, Lemma 3.2] or BenoistQuint [4, Lemma 4.36], we have a
	decomposition

  cm has µ ⊗d m -measure at least e -cm . By Theorem

	thm:LargeD 3.10 it:LargeDsv (ii) and Lemma	lm:place 5.4 below, if c
	is chosen small enough, there must exist (g 1 , . . . , g d ) satisfying this inequality, and
	moreover
	normgi normgi

Acknowledgements. We are indebted to Emmanuel Breuillard for several ideas used in § §3.2 and 3.3 and for sharing his unpublished note Breuillard [15] on non-concentration estimates for random matrix products. It is a pleasure to thank him, as well as Richard Aoun, Yves Benoist, Elon Lindenstrauss and Péter Varjú, for useful and motivating discussions.

W.H. is supported by ERC grant ErgComNum 682150.

Set η = c 20dλ1 , ρ e 20dλ1m , and for B = B(x, ρ), write e -cm ρ η ≤ (µ n * ν)(B) 2d

This shows that the µ ⊗2d m -measure of 2d-tuples of elements g 1 , . . . , h d such that

is at least e -cm . In particular, using the large deviation estimate Theorem thm:LargeD

3.10

it:LargeDn (i) and the observation above on the determinant, we may find elements g 1 , . . . , h d in the support of µ m satisfying

Now, the matrix g 1 + • • • -h d has integer entries, and its determinant is bounded above by e dλ1m , so that the entries of (g 1 + • • • -h d ) -1 are rational numbers with denominator bounded above by e dλ1m ≤ ρ -1 10 . Moreover, (g

. Equality ( ygv 5.1) above shows that x = g 1 y mod Z d is at distance at most ρ 9 10 from a rational point with denominator at most ρ - 1 10 . This finishes the proof.

Bootstrapping concentration.

We now wish to combine the diophantine property of µ n * ν with the concentration statement given by Proposition pr:step1

4.1 to obtain some concentration near rational points. To help the reader follow our progress towards Proposition pr:general 5.7, we formulate another intermediate step, which is the goal of this paragraph.

Given a subset X ⊂ T d and a small parameter ρ > 0, we shall write X (ρ) for the ρ-neighborhood of X. Thus, for instance, we write W (ρ) Q for the set of points in T d that lie at distance at most ρ from a rational point with denominator at most Q.

pr:step2 Proposition 5.2 (Second step: concentration around rational points). Under the assumptions recalled at the beginning of this section, there exists a constant C depending only on µ such that the following holds. Let t 0 ∈ (0, 1/2). Assume that for some

Then, writing W Q for the set of rational points on T d with denominator at most Q, and W (ρ) Q for its ρ-neighborhood, one has for every integer m such that m ≥ C|log t 0 | Then,

We are finally ready to prove the main theorem of this article, Theorem thm:main 1.2, announced in the introduction. We shall in fact prove a slightly more general statement, given as Proposition pr:general 5.7 below. Recall that for parameters Q ≥ 1 and ρ > 0, we write W Q for the set of rational points on T d with denominator at most Q, and W (ρ) Q for its ρ-neighborhood. ) ≥ t C .

Proof. In order to simplify notation, for any natural integer n, we write ν n = µ n * ν.

By Proposition

pr:step2

5.2, we may fix some m 0 large enough, so that there exists Q ∈ [e m0/C , e Cm0 ] such that

and apply repeatedly Proposition pr:rewindbis 5.5. After i steps, having obtained integers m 1 , . . . , m i-1 and ρ i > 0 such that

choose m i maximal so that e dλ1mi ρ i < Q -2 , and set ρ i+1 = e -λmi ρ i , so that by Proposition pr:rewindbis 5.5,

The procedure stops when nj m j < m * , the constant given by Proposition Email address: weikun.he@mail.huji.ac.il CNRS -Université Paris 13, LAGA, 93430 Villetaneuse, France.

Email address: desaxce@math.univ-paris13.fr