Dmitry Kleinbock
email: kleinboc@brandeis.edu

Nicolas De Saxcé
email: desaxce@math.univ-paris13.fr

Rational approximation on quadrics: a simplex lemma and its consequences

We give elementary proofs of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces X ⊂ P n (R). The main tool is a refinement of the simplex lemma, which essentially says that rational points on X which are sufficiently close to each other must lie on a totally isotropic rational subspace of X.

Introduction

The classical theory of Diophantine approximation studies the way points x ∈ R n are approximated by rational points p q ∈ Q n , taking into account the tradeoff between the size of q and the distance between p q and x; see cassels, schmidt [START_REF] Cassels | An introduction to Diophantine approximation[END_REF][START_REF] Schmidt | Diophantine approximation[END_REF] for a general introduction. Sometimes x is assumed to lie on a certain subset of R n , for example a smooth manifold X; this leads to the theory of Diophantine approximation on manifolds, in which there is no distinction between rational points which do or do not lie in X (this is referred to as ambient approximation).

Let now X be a rational quadric hypersurface of R n , let x ∈ X and let p q ∈ Q n be such that the distance between x and p q is less than ψ(q), where ψ is decaying fast enough, namely lim t→∞ t 2 ψ(t) = 0. Then p q must lie on X whenever q is large enough! This elementary observation, due to Dickinson and Dodson dickinsondodson [START_REF] Dickinson | Simultaneous Diophantine approximation on the circle and Hausdorff dimension[END_REF] for n = 2 and more generally to Druţu, see drutu [START_REF] Druţu | Diophantine approximation on rational quadrics[END_REF]Lemma 4.1.1], has in part motivated a new field of intrinsic approximation, which examines the quality to which points on a manifold are approximated by rational points lying on that same manifold. The paper kleinbockmerrill [START_REF] Kleinbock | Rational approximation on spheres[END_REF] studies the case X = S n-1 , the unit sphere in R n . Later in fkmsquadric [START_REF] Fishman | Intrinsic diophantine approximation on quadric hypersurfaces[END_REF] the results of kleinbockmerrill [START_REF] Kleinbock | Rational approximation on spheres[END_REF] were significantly strengthened and extended to the case of X being an arbitrary rational quadric hypersurface. An even more general framework was developed in fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]. Roughly speaking, in order to exhibit points on submanifolds X ⊂ R n which are close enough to rational points of X, one has to make use of the structure of X (indeed, in general it is not even guaranteed that X ∩ Q n is not empty). On the other hand, it is shown in fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF] that to prove some negative results, that is, to show that many points of X are not too close to rational points, one often does not need to know much about X. The main tool on which the argument of fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF] is based is the Simplex Lemma originating in Davenport's work davenport [START_REF] Davenport | A note on Diophantine approximation[END_REF]. The version presented in fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]Lemma 4.1] is very general -it applies to any manifold embedded in R nand at the same time precise enough to yield some satisfying theorems in the case of quadric hypersurfaces.

The purpose of this note is to show that in the special case where X is a rational quadric hypersurface, one can give more elementary and more geometric proofs of the results of fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]. This new approach will also yield more precise theorems. The main point is that one can prove a version of the simplex lemma with arbitrary hyperplanes replaced by Q-isotropic subspaces of X; this, in turn, yields refined information on the diophantine properties of X.

A detailed account of the results that are derived here is given in the next section. After that, in § sec:simplex 3 we prove the simplex lemma for quadrics, Lemma Acknowledgements. The authors are grateful to Emmanuel Breuillard, Nikolay Moshchevitin and Barak Weiss for motivating discussions.

2 General setting and main results of the paper exposition Since it will make the proofs more transparent, we shall from now on always work in the projective setting. We denote by P n (R) the n-dimensional real projective space. The natural map from R n+1 to P n (R) will be denoted by x → [x]. We now endow R n+1 with the standard Euclidean norm • , and explain how this defines a distance on P n (R). The distance between two elements x and y in P n (R) is equal to the sine of the angle between the two lines in R n+1 :

dist(x, y) := | sin(x, y)|. Equivalently, dist(x, y) = v x ∧ v y v x v y ,
where v x and v y are any nonzero vectors on x and y respectively, v x ∧ v y is the exterior product of v x and v y , and the Euclidean norm is naturally extended to ∧ 2 (R n+1 ) so that v x ∧ v y is the area of the parallelogram spanned by v x and v y .

If v = [v] ∈ P n (Q), where v = (v 1 , . . . , v n+1
) is an integer vector with coprime coordinates, the height of v is simply

H(v) := max 1≤i≤n+1 |v i |.
Given a point x in P n (R) we want to study how well x is approximated by points v in P n (Q).

Remark 2.1. In order to go back to the setting of Diophantine approximation in R n , one can consider an affine chart from an open subset of P n (R) to R n+1 . For example, if U = {[(x 1 , . . . , x n+1 )] : x n+1 = 0}, one can use the chart

U → R n , [(x 1 , . . . , x n+1 )] → ( x1 xn+1 , . . . , xn xn+1 ).
We consider a projective rational quadric X, given as the set of zeros of a rational quadratic form Q in n + 1 variables. Namely, for such Q let us consider

X = [Q -1 (0)] = x ∈ P n (R) : x = [x] with Q(x) = 0 .
(2.1) eq:X

Let us say that a subspace E ⊂ R n+1 is totally isotropic if Q| E ≡ 0. If E is as above, the projection [E] ⊂ X of E onto P n (R) will be referred to as a totally isotropic projective subspace. Recall that the Q-rank rk Q X of the quadric X is the maximal dimension of a totally isotropic rational subspace of R n+1 . If rk Q X > 0, this is the same as the maximal dimension of a totally isotropic rational projective subspace of X plus one. In particular, rk Q X > 0 if and only if X(Q) = ∅.

Given a point x in X, we shall be interested in the quality of rational approximations v ∈ X(Q) to x. The basic theory of such approximations has been developed in fkmsquadric [START_REF] Fishman | Intrinsic diophantine approximation on quadric hypersurfaces[END_REF]. In particular it was proved there

fkmsquadric [12, Theorem 5.1] that if rk Q X > 0 and X is nonsingular (2.
2) eq:nonsing (recall that a quadric hypersurface X is said to be nonsingular if the quadratic form that defines it is nondegenerate, i.e. has nonzero discriminant1 ), then for every x ∈ X there exists C x > 0 and a sequence

(v k ) ∞ 1 in X(Q) such that v k → x and dist(v k , x) ≤ C x H(v k ) . ( 2

.3) dirichlet

Thus if one defines the Diophantine exponent of x by

β(x) := inf β > 0 | ∃ c > 0 : ∀ v ∈ X(Q), dist(x, v) ≥ cH(v) -β , (2.4
) eq:beta then it follows that under the assumption ( eq:nonsing 2.2), β(x) ≥ 1 for all x ∈ X. On the other hand, it is shown in fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]Theorem 1.5] that the opposite inequality β(x) ≤ 1 is true for Lebesgue-almost every x ∈ X in the generality when X is not just a rational quadric but an arbitrary non-degenerate hypersurface. Moreover, the same is true if the Lebesgue measure is replaced by an absolutely decaying measure (see § sec:diophantine1 4.1 for definitions and more detail). This naturally leads to a question of exhibiting other measures µ on X such that β(x) ≤ 1 for µ-almost all x ∈ X. This is reminiscent to the subject of Diophantine approximation on manifolds and fractals, which has been extensively developed during recent decades for ambient approximation in R n , see bernikdodson [START_REF] Bernik | Metric Diophantine approximation on manifolds[END_REF], kleinbock-margulis [START_REF] Kleinbock | Flows on homogeneous spaces and Diophantine approximation on manifolds[END_REF] and

KLW

[16], for example. Measures satisfying the above property are usually called extremal. We shall also say that a submanifold Y ⊂ X is extremal if so is the Lebesgue measure on Y (by which we mean the restriction to Y of the k-dimensional Hausdorff measure where k = dim Y ).

Our first theorem, which is actually a special case of a more general result, Theorem Theorem 2.2 (Extremality of submanifolds of large dimension). Let X be a rational quadric hypersurface in P n (R), and let Y be a smooth submanifold of X with dim Y ≥ rk Q X. Then β(x) ≤ 1 for Lebesgue-almost every x ∈ Y .

In the case where X has Q-rank one, the above theorem provides a very simple and satisfactory answer to the problem of Diophantine approximation on submanifolds of X: any positive-dimensional submanifold Y ⊂ X is extremal. Note that there is no non-degeneracy condition on the submanifold Y . This comes in contrast to the case of approximation in R n , where one has to require that the submanifold is not included in an affine subspace.

In view of Theorem exi 2.2, it is natural to ask, given a submanifold Y of X of dimension at least rk Q X and a fixed β > 1, how large the intersection Y ∩ W β can be, where W β denotes the set of points in X whose Diophantine exponent is at least β. Note that it was proved in fkmsquadric [START_REF] Fishman | Intrinsic diophantine approximation on quadric hypersurfaces[END_REF]Theorem 6.4] that whenever X satisfies ( eq:nonsing 2.2), the Hausdorff dimension of W β is equal to n-1 β . Also in fishmanmerrillsimmons [START_REF] Fishman | Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces[END_REF] some upper estimates for the Hausdorff dimension of Y ∩ W β were obtained in the case when Y supports an absolutely decaying and Ahlfors-regular measure (see § sec:diophantine2

4.2 for details). Our second application of the simplex lemma strengthens the main result of fishmanmerrillsimmons [START_REF] Fishman | Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces[END_REF]. Here is a special case of a more general result, Theorem bei Theorem 2.3 (β-approximable points on submanifolds of large dimension). Let X be a rational quadric hypersurface in P n (R), and let

Y be a k-dimensional smooth submanifold of X with k ≥ rk Q X. Then one has dim H (Y ∩ W β ) ≤ k -(k + 1 -rk Q X)(1 -1 β ).
As the third application of our simplex lemma, we study the winning property of the set BA X of badly approximable points on X. Schmidt introduced games in his landmark paper schmidt_games [START_REF] Schmidt | On badly approximable numbers and certain games[END_REF] in order to study the set of badly approximable numbers in R n . He defined a winning property for subsets of R n , and showed the following:

• Any countable intersection of winning sets is winning ;

• If S is winning and f : R n → R n is a C 1 -diffeomorphism, then f (S) is winning; • If S ⊂ R n is winning then it has Hausdorff dimension n.
Then, Schmidt also showed that the set of badly approximable numbers in R n is winning. Variants of the Schmidt game were subsequently studied in numerous papers, among which bfkrw [START_REF] Broderick | The set of badly approximable vectors is strongly C 1 incompressible[END_REF] is the most relevant for the present purposes. In our setting, the set of badly approximable points on the quadric X is

BA X := {x ∈ X | ∃ c > 0 : ∀ v ∈ X(Q), dist(x, v) ≥ cH(v) -1 }.
(2.5) eq:defba

We define in § sec:diophantine3

4.3 a version of Schmidt's game, and show the associated winning property for the set BA X . As a corollary of this isotropically winning property, we get the following. bai Theorem 2.4 (Thickness of BA X on submanifolds of large dimension). Let X be a rational quadric hypersurface in P n (R). Then for any C

1 submanifold Y ⊂ X of dimension at least rk Q X, dim H (BA X ∩ Y ) = dim Y.
The properties of the set BA X have been studied in fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]. In particular, it was shown fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]Theorem 4.3] that BA X is hyperplane absolute winning (see § sec:diophantine3 4.3 for the definition and more detail); this gave the conclusion of the above theorem for Y = X. The refined version given above has the advantage that it is optimal: indeed, if Y is any totally isotropic rational projective subspace of

X of dimension rk Q X -1, then BA X ∩ Y = ∅.
3 Diagonal flows and the simplex lemma sec:simplex

The purpose of this section is to derive a simplex lemma, Lemma simplexquadric 3.1, for rational points on a rational quadric hypersurface X ⊂ R n+1 . For the proof, we shall relate good rational approximations to x ∈ X to the behavior of some diagonal orbit in the space of lattices in R n+1 .

Recall that the classical simplex lemma states that for each n ∈ N there exists c = c(n) > 0 such that if x is a point in R n and ρ ∈ (0, 1), then there exists an affine hyperplane containing all rational points with denominator at most cρ -n n+1 inside the ball B(x, ρ). The proof is based on the observation that any affinely independent n + 1 rational points with denominators at most D define inside B(x, ρ) a simplex whose volume can be bounded below by

1 n!D n+1 .
Therefore, one must have

1 n!D n+1 ≤ Vol B(x, ρ) = v n ρ n , where v n is the volume of the unit ball in R n , and hence D ≥ (n!v n ) -1 n+1 ρ -n n+1
. For a detailed proof, we refer the reader to kristensenthornvelani [20, Lemma 4]. The simplex appearing in the proof gave its name to the lemma.

Here we consider a rational quadratic form Q on R d and study rational points on X as in ( eq:X 2.1). simplexquadric Lemma 3.1 (Simplex lemma for quadric hypersurfaces). Let X be a rational quadric hypersurface in P n (R). Then there exists c > 0 such that for every ball B ρ ⊂ X of radius ρ ∈ (0, 1) the set

B ρ ∩ {v ∈ X(Q) | H(v) ≤ cρ -1 }
is contained in a totally isotropic rational projective subspace of X.

Let F Q be the symmetric bilinear form associated to the quadratic form Q defining X. The kernel of Q is defined by

ker Q = {x = [x] ∈ P n (R) | ∀ y ∈ R n+1 , F Q (x, y) = 0}. Assuming that X(Q) ker Q is non-empty, we may write, in some rational basis of R n+1 , Q(x 1 , . . . , x n+1 ) = 2x 1 x n+1 + Q(x 2 , . . . , x n ), (3.1) qgoodform
where Q is a quadratic form in n -1 variables. Let G = SO Q (R) be the group of unimodular linear transformations of R n+1 preserving the quadratic form Q.

The group G acts transitively on X ker Q, which may be identified with the quotient space X P \G, where P is the stabilizer of the isotropic line [e 1 ] in the standard representation. In fact, for x ∈ X ker Q, we may choose

u x ∈ G ∩ O n+1 (R) such that u x x = [e 1 ].
We shall consider the diagonal subgroup a t = diag(e -t , 1, . . . , 1, e t ) in G, and if x ∈ X, let

g x t = u -1 x a t u x .
The lemma below is due to Kleinbock-Merrill kleinbockmerrill [START_REF] Kleinbock | Rational approximation on spheres[END_REF] in the case of projective spheres, and to Fishman-Kleinbock-Merrill-Simmons fkmsquadric [START_REF] Fishman | Intrinsic diophantine approximation on quadric hypersurfaces[END_REF]Lemma 7.1] in the general case. To make the paper self-contained, we provide a proof here. daniquadric Lemma 3.2 (Dani correspondence for quadric hypersurfaces). Let Q be as in ( qgoodform 3.1), and write X for the associated rational quadric hypersurface in P n (R). With the above notation, there exists C > 0 such that for x ∈ X and v ∈ X, we have, for all t ∈ R,

g x t v ≤ C max(e -t H(v), H(v) dist(x, v), e t H(v) dist(x, v) 2 ),
where v ∈ Z n+1 is a representative of v with coprime integer coordinates.

Proof.

Fix C 0 ≥ 2 larger than max w =1 | Q(w)|, so that for all w in R n-1 , | Q(w)| ≤ C 0 w 2 .
With u x as above, write

u x v = v 1 e 1 + v 2 e 2 + • • • + v n+1 e n+1 .
Letting w = v 2 e 2 + • • • + v n e n , we have

u x g x t v = e -t v 1 e 1 + w + e t v n+1 e n+1 ,
and therefore, since u x is in O n+1 (R),

g x t v ≤ 3 max(e -t |v 1 |, w , e t |v n+1 |). (3.2) gtxv Now note that |v 1 | ≤ H(v) and H(v) ≥ 1 √ n+1 v , so √ n + 1H(v) dist(x, v) ≥ u -1 x e 1 ∧ v = e 1 ∧ u x v = e 1 ∧ (w + v n+1 e n+1 ) = w + v n+1 e n+1 ≥ w . Moreover, Q(u x v) = 0 yields |v n+1 | = | Q(w)| 2|v 1 | ≤ C 0 w 2 2|v 1 | , so that, provided dist(x, v) ≤ √ 2 2 , |v n+1 | ≤ C 0 2
H(v) dist(x, v) 2 1 -dist(x,v) 2 H(v) 2 ≤ C 0 H(v) dist(x, v) 2 . Of course, if dist(x, v) ≥ √ 2
2 , we also have

|v n+1 | ≤ H(v) ≤ C 0 H(v) dist(x, v) 2 , because C 0 ≥ 2. Going back to ( gtxv 3.
2), we find the desired inequality, with C = max(3C 0 , √ n + 1).

We can now prove the simplex lemma.

Proof of Lemma simplexquadric 3.1. Let Q be a quadratic form defining the hypersurface X. The result is obvious if X(Q) ⊂ ker Q, so we may assume that X(Q) ker Q is non-empty. Then, replacing Q if necessary by an integer multiple, we may find an integer basis of R n+1 in which Q has the form (

qgoodform 3.1). Let C 1 = max v =1 |Q(v)|, so that for all v ∈ R n+1 , |Q(v)| ≤ C 1 v 2 , and let c = 1 C √ 5C1
, where C is the constant given by Lemma daniquadric 3.2. We need to show that any family v 1 , . . . , v s of points in X(Q) ∩ B(x, ρ) satisfying H(v i ) ≤ cρ -1 , i = 1, . . . , s, generates a totally isotropic subspace. For each v i , we take a representant v i in Z n+1 with coprime integer coordinates. It is enough to show that for all i and j, Q(v i ±v j ) = 0, and since the quadratic form Q takes integer values at integer points, it suffices to check that for all i and j, |Q(v i ± v j )| is less than 1.

Now, choosing t > 0 such that e t = ρ -1 , Lemma daniquadric 3.2 shows that g x t v i ≤ Cc. Then, we write

Q(v i ± v j ) = Q(g x t v i ± g x t v j )) ≤ C 1 g x t v i ± g x t v j 2 ≤ 4C 1 (Cc) 2 = 4 5 .
This implies what we want.

spheres Remark 3.3. In the case when X = S n-1 is the (n -1)-dimensional sphere, identified with the subset of R n defined by the equation

x 2 1 + • • • + x 2 n = 1
, one can give a more direct proof of the simplex lemma. Indeed, if p1 q1 and p2 q2 are two distinct rational points on S n-1 of height at most ρ -1 2 , we have

p 1 q 1 - p 2 q 2 2 = 2 - p 1 • p 2 q 1 q 2 ≥ 1 q 1 q 2 ≥ 4ρ 2 ,
so that any open ball of radius ρ contains at most one rational point of height at most ρ -1 2 . In fact, such a direct computation can also be made for a general quadric hypersurface, but we chose to give a more geometric proof of Lemma Remark 3.4. When the quadratic form Q has Q-rank one, the only isotropic rational projective subspaces are points in X(Q). This makes the consequences of the simplex lemma more spectacular in the particular case of Q-rank one.

Applications to Diophantine approximation sec:diophantine

In this section, as before, X is a rational quadric hypersurface in P n (R) defined by a rational quadratic form Q. We are concerned with intrinsic Diophantine approximation on X, which is the study of the quality of approximations of a point x in X by rational points v lying on X. On that matter, the simplex lemma has several simple consequences, which we now explain.

Extremality sec:diophantine1

Recall that the Diophantine exponent of a point x ∈ X was defined by ( eq:beta 2.4). Our next theorem generalizes Theorem exi 2.2 using the following definition.

def-ad

Definition 4.1. Given a positive parameter α, a finite Borel measure µ on the quadric hypersurface X will be called α-isotropically absolutely decaying, abbreviated as α-IAD, if there exists a constant C > 0 such that for every x ∈ X and every totally isotropic rational projective subspace L ⊂ X, ∀ ε > 0 ∀ ρ ∈ (0, 1), µ B(x, ρ) ∩ L (ερ) ≤ Cε α µ B(x, ρ) , (4.1) eq:ad where L (τ ) denotes the neighborhood of size τ of the set L. We shall say that µ is isotropically absolutely decaying (IAD) if it is α-IAD for some α > 0.

extremalquadric Theorem 4.2 (IAD measures are extremal). Let X be a rational quadric in P n (R), and let µ be an IAD measure on X. Then β(x) ≤ 1 for µ-almost every x ∈ X.

Remark 4.3. Recall that a measure µ is called α-absolutely decaying if ( eq:ad 4.1) holds for some C > 0, every x ∈ X and every subspace L ⊂ P n (R), and absolutely decaying if it is α-absolutely decaying for some α > 0. It follows from fkmsgeneral [13, Theorem 1.5] that for any absolutely decaying measure µ on X one has β(x) ≤ 1 for µ-almost every x ∈ X. In fact it holds more generally when X is not just a rational quadric but an arbitrary non-degenerate smooth hypersurface.

Absolutely decaying measures are IAD but not vice versa. In particular, the Lebesgue measure on a smooth proper submanifold Y of X with dim Y ≥ rk Q X is not absolutely decaying but α-IAD with α = dim Y -rk Q X + 1; so Theorem [START_REF] Weiss | Almost no points on a cantor set are very well approximable[END_REF] for a one-dimensional version. By the Borel-Cantelli lemma, it is enough to check that for all ε > 0,

Proof of Theorem

k≥1 µ x ∈ X ∃ v ∈ X(Q) : 2 k ≤ H(v) < 2 k+1 dist(x, v) ≤ 2 -k(1+ε) < ∞.
Fix k ≥ 1. There exists an integer K such that we may cover X by a family of balls

B i = B(x i , 2 -k(1+ ε 2 )
), i = 1, . . . , N , so that any intersection of more than K distinct balls is empty. By Lemma simplexquadric 3.1, for k large enough, for each i, the set of points v ∈ X(Q) ∩ B i satisfying 2 k ≤ H(v) < 2 k+1 is contained in a totally isotropic rational subspace L i , and therefore, by the IAD property of µ for some C, α > 0 one has

µ x ∈ B i ∃ v ∈ X(Q) : 2 k ≤ H(v) < 2 k+1 dist(x, v) ≤ 2 -k(1+ε) ≤ µ B i ∩ L (2 -k(1+ε) ) i ≤ C2 -kα ε 2 µ(B i ).
Summing over all balls B i , and using the fact that the cover (B i ) i∈N has multiplicity at most K, we get

µ x ∈ X ∃ v ∈ X(Q) : 2 k ≤ H(v) < 2 k+1 dist(x, v) ≤ 2 -k(1+ε) ≤ KC2 -kα ε 2 .
Since this last bound is summable in k, this concludes the proof of the theorem. 4.1 are zero-dimensional, and isotropic absolute decay coincides with weak absolute decay as defined in bgsv [START_REF] Beresnevich | Diophantine approximation in Kleinian groups: singular, extremal and bad limit points[END_REF]. Moreover, in the case where X is a sphere, Theorem Remark 4.5. We could have stated a slightly stronger version of the theorem, in the form of a Khintchine-type theorem: if µ is α-IAD, and if ψ : R + → R + is a non-increasing function satisfying

k∈N k α-1 ψ(k) α < ∞, then for µ-almost every x in X, there exists c > 0 such that ∀v ∈ X(Q), dist(x, v) ≥ cψ H(v) .
The proof, based on the easy half of the Borel-Cantelli lemma, is essentially the same as the one presented above.

Hausdorff dimension and Diophantine exponents sec:diophantine2

As a complement to the above study of the extremality problem, we explain here how the simplex lemma can be used to give a simple proof of a recent result of Fishman-Merrill-Simmons fishmanmerrillsimmons [START_REF] Fishman | Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces[END_REF]. Once again, X denotes a rational quadric projective hypersurface of dimension n. Given β > 0, we shall be concerned with the set

W β = {x ∈ X | β(x) ≥ β}.
Given a subset K in X, our goal will be to bound the Hausdorff dimension of the intersection K ∩ W β ; we shall be able to do so if K is the support of a sufficiently regular measure. For δ > 0, a Borel measure µ on a metric space X is said to be Ahlfors-regular of dimension δ if we have, for some constant A > 0,

∀ x ∈ X ∀ r ∈ (0, 1], 1 A r δ ≤ µ B(x, r) ≤ Ar δ .
We now present a short proof of a strengthening of fishmanmerrillsimmons [14, Theorem 1.2], using Lemma simplexquadric 3.1. hausdorffexponent Theorem 4.6. Let X be a rational quadric projective hypersurface. Let µ be an Ahlfors-regular measure of dimension δ on X, and let K = Supp µ. If µ is α-IAD, then we have, for all β ≥ 1, . However in our decay condition we only have to consider totally isotropic subspaces. In particular, Theorem hausdorffexponent 4.6 covers the case where K is a smooth submanifold of X of dimension at least rk Q (X), and therefore generalizes Theorem Proof of Theorem hausdorffexponent 4.6. If β = 1, there is nothing to prove, so we assume β > 1 and fix γ ∈ (1, β). For p ≥ 0, let

dim H (K ∩ W β ) ≤ δ -α 1 - 1 β . ( 4
A p = x ∈ X ∃ v ∈ X(Q) : 2 p ≤ H(v) < 2 p+1 dist(x, v) ≤ 2 -γp .
Taking a maximal 2 -p -separated subset {x i } 1≤i≤ p of K ∩ A p , the collection of balls C p = B(x i , 2 -p ) 1≤i≤ p covers K ∩ A p and has multiplicity bounded above by some constant C depending only on X. Using the Ahlfors regularity of µ, this implies p 2 -pδ ≤ ACµ(X) = AC, i.e. p ≤ AC2 pδ .

Since γ > 1, Lemma simplexquadric 3.1 shows that for p large enough, for each ball B ∈ C p , there exists a totally isotropic subspace

L B of X such that A p ∩ B ⊂ L (2 -γp ) B
. So the decay condition on µ yields, up to multiplicative constants depending only on X and µ, that

µ(A p ∩ B) 2 -(γ-1)αp µ(B) 2 -p[δ+(γ-1)α] .
Next, take a minimal cover

D B = (B i ) i∈I B of the set K ∩ A p ∩ B by balls of radius 2 -γp centered on K ∩ A p ∩ B.
Just as above, the Ahlfors regularity of µ shows that

#I B 2 δγp µ(A p ∩ B) 2 pγδ 2 -p[δ+(γ-1)α] .
Thus, we find for every s > 0,

B∈Cp i∈I B (diam B i ) s 2 pδ 2 p(γ-1)(δ-α) 2 -pγs = 2 -p[sγ-γδ+α(γ-1)]
If s > δ -α 1 -1 γ , then the family of balls (B i ) i∈I B , B∈Cp, p∈N satisfies the assumption of the Hausdorff-Cantelli lemma, and therefore, letting

s → δ -α 1 - 1 γ we find that dim H (lim sup B i ) ≤ δ -α 1 -1 γ . Now, since γ < β, we have K ∩W β ⊂ (lim sup B i ), hence letting γ → β, we can conclude that the Hausdorff dimension of K ∩ W β is not greater than δ -α 1 -1 β .
In the case of Q-rank one, any Ahlfors-regular measure of dimension δ is δ-IAD, so we get the following corollary, which applies in particular when X = S n-1 is the unit sphere in R n : Corollary 4.9. Let X be a rational quadric hypersurface of Q-rank one, and let µ be an Ahlfors-regular measure of dimension δ on X. Writing K = Supp µ, we have, for every

β ≥ 1, dim H (K ∩ W β ) ≤ δ β .

Badly approximable points sec:diophantine3

Recall the definition ( eq:defba 2.5) of the set BA X of intrinsically badly approximable points in X. As was mentioned in Section exposition 2, it is known fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF] to satisfy some winning properties in the sense of Schmidt's games. Our goal will now be to give a more elementary proof of a refinement of the winning property, again using the simplex lemma.

We now explain the principles of our version of Schmidt's game. As before, X is a rational quadric hypersurface of P n (R). There are two players, Alice and Bob, and some parameter β ∈ (0, 1 3 ). To start, Bob chooses a ball B 0 = B(x 0 , ρ 0 ) in X. Then, at each stage of the game, after Bob has chosen a ball B i = B(x i , ρ i ), Alice chooses a totally isotropic rational subspace L of X and deletes its neighborhood of size ε, with 0 < ε ≤ βρ i .

A set S is isotropically β-winning if Alice can make sure that

B i ∩ S = ∅.
Finally, S is isotropically winning if it is isotropically β-winning for arbitrarily small β > 0. Our game is inspired by Broderick, Fishman, Kleinbock, Reich and Weiss bfkrw [START_REF] Broderick | The set of badly approximable vectors is strongly C 1 incompressible[END_REF], where the authors define the notion of k-dimensionally absolute winning using exactly the same game, except that Alice is allowed to delete neighborhoods of arbitrary k-dimensional subspaces. In particular, we have the following properties of isotropically winning sets. Proposition 4.10 (Properties of winning sets). Let X be a projective quadric hypersurface in P n (R).

1. If S is isotropically winning on X, then S is dense and dim H S = dim X. Remark 4.11. We warn the reader that the image of an isotropically winning set under a C 1 diffeomorphism of X may not be isotropically winning. However, by bfkrw [4, Proposition 2.3.(c)], it will certainly be k-dimensionally absolute winning, and therefore dense and with maximal Hausdorff dimension.

The following theorem is a refinement of fkmsgeneral [START_REF] Fishman | Intrinsic Diophantine approximation on manifolds: general theory[END_REF]Theorem 4.3]: winning Theorem 4.12 (Badly approximable points on X are winning). Let X be a rational quadric hypersurface in P n (R). Then the set BA X is isotropically winning.

Proof. Fix β ∈ (0, 1 3 ). Bob first picks a ball B 0 = B(x 0 , ρ 0 ). By Lemma simplexquadric 3.1, there exists a constant c > 0 depending only on X such that all rational points v in 2B 0 satisfying H(v) ≤ cρ -1 0 are included in some totally isotropic rational subspace L 0 . Alice deletes L (βρ0) 0

. Similarly, once Bob has chosen a ball B i = B(x i , ρ i ), the rational points v ∈ 2B i such that H(v) ≤ cρ -1 i all lie on a hyperplane L i , and Alice deletes L (βρi) i . If there is no rational point of small height in B i , then Alice can delete a ball of radius βρ i around the center. This ensures that ρ i → 0.

We claim that this strategy forces i≥0 B i ⊂ BA X . To see this, let x ∈ B i and v ∈ X(Q). Choose i such that

cρ -1 i-1 ≤ H(v) ≤ cρ -1 i . (4.3) rhoi If v ∈ 2B i , then, using x ∈ B i , we find dist(x, v) ≥ ρ i ≥ βρ i-1 ≥ βcH(v) -1 . And if v ∈ 2B i , then ( rhoi 4.3) implies that v ∈ L i , and since x ∈ B i+1 , dist(x, v) ≥ βρ i ≥ β 2 ρ i-1 ≥ β 2 cH(v) -1 .
Taking c 0 = cβ 2 , we find

∀ v ∈ X(Q), dist(x, v) ≥ c 0 H(v) -1 , so x ∈ BA X .
As is the case with the k-dimensional absolute game, the advantage of the isotropic game is the inheritance of winning properties to sufficiently regular subsets. More precisely, given a compact subset K ⊂ X, we may consider the isotropic game played on K. The rules are the same as before, but the ambient metric space is now K: at each stage, Bob chooses a ball B(x i , ρ i ) centered on K, and Alice deletes the intersection of K with the neighborhood of size βρ i of a rational isotropic subspace. Naturally, we shall say that a set S is isotropically winning on K if S ∩ K is winning for the isotropic game on K.

Following Broderick, Fishman, Kleinbock, Reich and Weiss bfkrw [START_REF] Broderick | The set of badly approximable vectors is strongly C 1 incompressible[END_REF], let us say that a subset K ⊂ X is isotropically diffuse if there exists β, ρ K > 0 such that for every ρ ∈ (0, ρ K ), x ∈ K, and every totally isotropic rational subspace L, the set K ∩ B(x, ρ) L (βρ) is non-empty. This is a quantitative way to say that K is nowhere included in a small neighborhood of a totally isotropic subspace. The next lemma is a straightforward analogue of bfkrw [4, Proposition 4.9].

diffuse Lemma 4.13. Let X be a rational quadric hypersurface in P n (X). If L ⊂ K are two isotropically diffuse subsets of X, and S ⊂ X is isotropically winning on K, then S is isotropically winning on L.

The proof is very similar to the one presented in bfkrw [START_REF] Broderick | The set of badly approximable vectors is strongly C 1 incompressible[END_REF], once one has replaced the notions of k-dimensionally diffuse and k-dimensionally winning by those of isotropically diffuse and isotropically winning. We refer the reader to bfkrw [4, Section 4] for details.

It follows from the above lemma and Theorem winning 4.12 that BA X is isotropically winning on any isotropically diffuse subset of X. This in particular applies to smooth submanifolds Y of X of dimension not less than rk Q (X), which are isotropically diffuse. Furthermore, the Lebesgue measure on Y as above is Ahlfors-regular of dimension equal to dim Y . Therefore, in view of Remark 4.14. In the case of X = S n-1 , or more generally of a rational quadric of Q-rank one, the above shows that BA X is winning on any positive-dimensional submanifold of X. This can be compared with a similar question for Diophantine approximation in Euclidean spaces, for which it is still open, despite recent progress of Beresnevich Singular points. Given a rational quadric X in P n (X), one may define, for c > 0,

D(c) =    x ∈ X ∃ N 0 : ∀ N ≥ N 0 ∃ v ∈ X(Q) such that H(v) ≤ N and dist(x, v) ≤ c N H(v)    ,
and call a point x ∈ X singular if x ∈ c>0 D(c). If X has Q-rank 1, it follows from Dani's work dani_divergent [START_REF] Dani | Divergent trajectories of flows on homogeneous spaces and Diophantine approximation[END_REF] that x is singular if and only if x ∈ X(Q). In fact, one can show that if X has Q-rank 1, D(c) = X(Q) for c > 0 small enough. This follows for example from the following strengthening of Lemma simplexquadric 3.1, whose proof is identical up to some minor changes. See also sumofsquares [START_REF] Kleinbock | Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials[END_REF]Theorem 3] for an alternative proof.

Lemma 5.1 (A stronger simplex lemma for quadric hypersurfaces). Let X be a rational quadric hypersurface in P n (R). Then there exists c > 0 such that, for every x ∈ X and any ρ ∈ (0, 1), the set

v ∈ X(Q) H(v) ≤ cρ -1 , dist(x, v) ≤ ρ H(v)
is contained in a totally isotropic rational subspace L ⊂ X.

When the quadric X has Q-rank at least 2, it is natural to expect that there exist some nontrivial singular points. It might then be interesting to compute the Hausdorff dimension of the set of singular points on X, similarly to what has been done in cheung,cheung-chevallier [START_REF] Cheung | Hausdorff dimension of the set of singular pairs[END_REF][START_REF] Cheung | Hausdorff dimension of singular vectors[END_REF] for Diophantine approximation in the Euclidean space.

Extremality. In view of the definitive results in the area of Diophantine approximation on manifolds and fractals obtained in kleinbock-margulis [START_REF] Kleinbock | Flows on homogeneous spaces and Diophantine approximation on manifolds[END_REF], it is natural to attempt to weaken the condition of isotropic absolute decay of µ as in Theorem extremalquadric 4.2, and conjecture that on a general quadric hypersurface, any analytic submanifold that is not included in an isotropic subspace is extremal. In fact, by analogy with kleinbock_inheritance [START_REF] Kleinbock | Extremal subspaces and their submanifolds[END_REF], one can guess that an analytic submanifold on a quadric hypersurface inherits its Diophantine exponent from the smallest totally isotropic subspace in which it is contained.

Other projective varieties. One may wonder how general is the approach presented here, and whether it can be used to study intrinsic Diophantine approximation on varieties that are not quadric hypersurfaces.

simplexquadric 3 . 1 ,

 31 which is central in all the subsequent developments. Applications of the simplex lemma to Diophantine approximation on quadrics are presented in § sec:diophantine 4. Those results are proved along the same lines as the analogous statements for Diophantine approximation in the Euclidean space R n , but the proofs are included to make the paper self-contained. Finally, in § sec:open 5 we discuss some open problems and possible further directions for the study of intrinsic Diophantine approximation on projective varieties.

[ 13 ,

 13 Theorem 1.5] for rational quadrics X as follows:

exi 2 .

 2 2 is a corollary from Theorem extremalquadric 4.2.

extremalquadric 4 . 2 .

 42 The argument follows the lines of the proof of PV [21, Theorem 1], see alsoW

Remark 4 . 4 .

 44 When rk Q (X) = 1, all the subspaces L appearing in Definition def-ad

hausdorffexponent 4 .

 4 6 is a straightforward adaptation of that of PV [21,Theorem 2]. We shall use the easy Hausdorff-Cantelli lemma stated below.

Lemma 4 . 8 ([ 3 ,

 483 Hausdorff-Cantelli). Let (B i ) i≥0 be a family of balls in a metric space, and assume that i≥0 (diamB i ) s < ∞. Then, dim H (lim sup B i ) ≤ s.Proof. Left as an exercise, see Bernik-Dodson bernikdodson Lemma 3.10].

2 .[ 4 ,

 24 If (S i ) i∈N is a countable family of isotropically winning sets on X, then i∈N S i is isotropically winning. Proof. Let k = rk X -1. Any isotropically winning set is k-dimensionally absolute winning in the sense of bfkrw [4, page 323], so that the first item follows from the analogous property for k-dimensional absolute winning bfkrw Proposition 2.3]. Alternatively, one may adapt the proof of Schmidt schmidt_games [22, Theorem 2]. The proof of the second item is identical to the analogous statement for k-dimensional absolute winning, see bfkrw [4].

bfkrw [ 4 ,

 4 Lemma 5.3], for every open subsetU of X such that U ∩ Y = ∅, one has dim H (Y ∩ BA X ∩ U ) = dim Y, which implies Theorem bai 2.4.

5

  Further directions and open problems sec:open Khintchine's theorem. It would be interesting to use the geometric observations of this note to give an elementary proof of Khintchine's theorem on quadric hypersurfaces, due to Fishman, Kleinbock, Merrill and Simmons fkmsquadric

This is also equivalent to X being nonsingular as a projective algebraic variety.

* D.K. was supported in part by NSF grant DMS-1600814.


T F Boukezoula

L Bencheikh

D E C Belkhiat

The impact of non-uniform bifunctional catalyst for the isomerization of n-hexane in fixed bed catalytic reactor

Keywords: bifunctional catalyst, non-uniform distribution, activity, selectivity

Catalytic reforming is a very important process in chemical industry. The aim of this process is to transform treated naphtha (paraffins, olefins, naphtens, aromatics) into reformates with a better yieldin aromatics such as benzene, toluene, and xylenes (BTX) that are very important petrochemical materials and providing a high octane index. In this work, a mathematical model is proposed to investigate the influence of the concentration and the non-uniform distribution of the catalytic sites on the performances of an isothermal fixed bed reactor with axial dispersion and mass transfer resistance. This model is applied to the naphtha catalytic reforming network. The simulation of the model for a particular chemical reaction network showed that the catalyst activity and selectivity depend on the balance between the number, per unit volume, of the active sites and the way they are distributed throughout the pellet.

INTRODUCTION

It is well established that the presence of poisoning and deactivation phenomena by coke deposit on the external lateral surface of the catalytic pellet reduces activity and selectivity and stability of the bi-functional catalytic pellet. The performance of this latter is conditioned by the way the catalytic sites are distributed inside the pellet. It is here that lies the importance of studying the active site distribution inside the catalytic pellet. It has long been known that a uniform distribution is not always the adequate choice for an optimal functioning. This is particular important when using noble and costly materials as catalysts. An optimal use of these materials is thus necessary.

II. LITERATURE REVIEW

Dario et al. [START_REF] Dario | Activity and selectivity of Non uniform bifunctional catalysts[END_REF][START_REF] Dario | Activity and selectivity of non uniform bifunctional catalysts: Analysis of the fixed -bed reactor performance[END_REF] studied theoretically the influence of the distribution profiles of the active sites on both activity and selectivity of a bi-functional non uniformly distributed catalyst, in a catalytic cylindrical particle and in the fixed bed reactor without axial dispersion. They established that catalytic activity and selectivity of the different species were largely influenced by the active site concentration and distribution when searching optimal performances of the catalyst. Morbidelli et al. [START_REF] Morbidelli | Optimal catalyst activity profiles in pellets .1. The case of negligible external mass transfer resistance[END_REF][START_REF] Morbidelli | Optimal catalyst activity profiles in pellets. 2. The influence of external mass transfer resistance[END_REF][START_REF] Chemburkar | Optimal catalyst activity profiles in pellets-VII. The case of arbitrary reaction kinetics with finite external heat and mass transport resistance[END_REF], showed analytically that for a given amount of active metal, the Dirac distribution, in which the catalyst was concentrated at a specified position, leads to an optimal efficiency. Gavriilidis and Varma [START_REF] Gavriilidis | Optimal distribution of catalyst in pellets[END_REF] analyzed numerically the problem of practically positioning the active material. The active sites were distributed in a narrow zone (thin layer) around the optimal position, with a thickness less than 5% of the catalytic particle diameter. This optimal position, which depends on the Thiele modulus, corresponds to a maximum reaction rate.

III. MATHEMATICAL FORMULATION OF THE PROBLEM

A. Kinetic model

Let us consider a complex reaction scheme involving ' S ' species which can be represented by R reaction series.

  , 1, 2,...... ij k ij A A i j S i j     (1)
In the case of a bifunctional catalytic particle, with two types of active sites (site I and site II) existing simultaneously, the reaction rate is supposed to be written as the sum of the two contributions, that is:

    ij ij I ij II i r k x k x c       (2)
The active sites distribution is represented by the function   x  .

B. Dimensionless equations

Before solving these equations, they need to be converted into a dimensionless form. Let us introduce the following dimension less variables:

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x a

  , z L   , 10 i i u c  , 10 i i u U u  (3) 1) Mass balance in catalyst phase 2 22 2 1 S ii ij i ji j j ii ji dd p a a d D D d              (4)   0 , 1 0 i i i i s d d d Bio U d                    (5)       1 S ij ij I ij II j ij kk                   (6)       ji ji I ji II kk                (7)
2) Mass balance in fluid phase

  2 * , 2 1 0 ii L i i s d U dU KU Pe d d        (8)     1 0 0 0 1 1 1 01 i i i i dU d U for i dU U for i Pe d                        (9)
Where

Nature of      Mathematical expression Variation Parabolic -2     2 61 s C     decreasing +2   2 2 s C   increasing Linear -1     31 s C     decreasing +1   1.5 s C   increasing Uniform 0   s C   Constant
The  parameter is used to identify the nature of the distribution profile.

IV. APPLICATION

In order to simulate our mathematical model, it was tested on a reaction scheme involving the catalytic isomerization of hexane (see fig. 1).

Two kinds of reactions appear in this reaction network:  The reactions taking place on acid sites are represented horizontally (Site I).  The reactions taking place on metallic sites are represented vertically (Site II). Fig. 2 shows that the concentration profiles of all species, along the catalytic reactor, are very affected by sII C . It should be noted that the concentration of species 1 A decreases rapidly with higher values of sII C . This is probably due to the fact that most active metallic sites are concentrated at the inner peripheral surface of the particle. The concentration of the

Site II

Site (I)

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desired species 4 A also seems to increase with higher values of sII C .

 II =2  II =1  II =0  II =-2  II =-1 P=1 Pe=1000 Bio=100 =25s Cs I =1,Cs II =1  I =0 a2) U 2   II =2  II =1  II =0  II =-2  II =-1 P=1 Pe=1000 Bio=100 =25s Cs I =1,Cs II =1  I =0 a3) U 3   II =2  II =1  II =0  II =-2  II =-1 P=1 Pe=1000 Bio=100 0,1 0,2 0,3 0,4 0,5 0,6 0,7 a1) U 1 
Cs I =1,Cs II =1  I =0 a4) U 4   II =2  II =1  II =0  II =-2  II =-1 P=1 Pe=1000 Bio=100 =25s
Cs I =1,Cs II =1  I =0 =25s

VII. CONCLUSION

In order to improve the catalytic performances in terms of efficiency and selectivity with respect to the desired species 4 A , a mathematical model, based on an isothermal fixed bed, reactor hypothesis, was proposed and simulated. The results obtained for the concentration profiles of the active species led to the following; the concentrations of active sites and their distributions. In order to obtain a high conversion rate, the metallic sites involved in the first reaction, i.e.

K

  : is a dimension less transfer coefficient (fluid-solid) C. Choice of the distribution function The mathematical expressions of the different distribution functions used are summarized in the following table:

Fig. 1 .

 1 Fig. 1. Catalytic isomerization of hexane.

Fig. 2 .

 2 Fig. 2. Effect of   sII C

Fig. 3 .

 3 Fig. 3. Effect of the metallic site distribution on the concentration profiles of all species along the reactor

a

    ) must be concentrated at the inner peripheral surface of the catalytic particles.It was shown in this work that the mathematical model proposed can be applied to any chemical reaction network and an appropriate choice of the different parameters of the problem can led to optimum results.Indicates the half thickness the catalytic pelletThe International Conference on Advanced Engineering in Petrochemical Industry (ICAEPI'

TABLE I .

 I

CONTINUOUS DISTRIBUTION FUNCTIONS

[START_REF] Morbidelli | Optimal catalyst activity profiles in pellets. 2. The influence of external mass transfer resistance[END_REF]