G Tenenbaum
email: gerald.tenenbaum@iecn.u-nancy.fr

M Tucsnak
email: tucsnak@iecn.u-nancy.fr

On the null-controllability of diffusion equations *

Introduction and main results

This work is concerned with the study of null-controllability for a class of infinite dimensional systems described by abstract parabolic equations of the form

(1.1) ẇ = Aw + Bu, w(0) = z,
where A is a negative operator in a Hilbert space X and, say, the operator B maps the input space U (also a Hilbert space) into X. As discussed below, the range of B can in fact be a space which is larger than X: we say in this case that the input operator in unbounded. More precise assumptions on A and B will be made later.

In the special case when A is the Dirichlet Laplacian in an open bounded set Ω ⊂ R n and B corresponds to a control distributed in a subset of Ω, i.e., for the standard diffusion equation, the null-controllability properties are well understood. For early contributions, in which A is the Dirichlet Laplacian in one space dimension, we refer to Fattorini and Russell [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF], [START_REF]Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF].

When A is the Dirichlet Laplacian in Ω ⊂ R n , where n 2, and the control u is supported on an arbitrary open subset of Ω, the null-controllability result has been independently established by Lebeau and Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], and by Fursikov and Imanuvilov [START_REF] Fursikov | Controllability of Evolution Equations[END_REF].

More recently, Micu and Zuazua in [START_REF] Micu | On the controllability of a fractional order parabolic equation[END_REF] (in one space dimension) and Miller [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] (in several space dimensions) obtained a null-controllability result when A = -A η 0 , where η > 1 2 and -A 0 is the Dirichlet Laplacian. The main tool brought up in [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] is an abstract sufficient condition for null-controllability, inspired by the method introduced in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]-see also Lebeau and Zuazua [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF]. Note that the spectral condition in [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] is stated for a A = -A η 0 , with a general positive operator A 0 (not necessarily with discrete spectrum) and that it furnishes estimates of the control cost. The work of Seidman [START_REF] Seidman | How violent are fast controls. III[END_REF] contains a new version of the Lebeau-Robbiano method which allowed, at least for η = 1 and B ∈ L (U, X), the improvement of the estimates from [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF]. The constants in the control cost have been further improved in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF], where a more direct methodology has been proposed.

In this work, our initial aim consists, in an abstract setting, in relaxing the assumptions on the control operator B with respect to the existing literature, together with simplifying some proofs and the form of the constants involved in the control cost estimates. In this context, without assuming any admissibility property of the control operator B, we show, without any duality argument, that the observability assumption of finite combinations of eigenvectors of the pair (A * , B * ) implies its final-state observability. Note that a proof of this implication has also been, independently and almost simultaneously, provided in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] under the assumption that B is an admissible observation operator for the semigroup generated by A.

Our second purpose is, in the case when A = -A η 0 , where A 0 is the Dirichlet Laplacian in a rectangular domain Ω and η > 1/2, to provide an alternative way to check the Lebeau-Robbiano spectral condition. We show, in this particular case, that the sophisticated Carleman and interpolation inequalities used in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] may be replaced by a simple result of Turán [START_REF] Turán | On a theorem of Littlewood[END_REF]. A novelty brought in by our approach is that, in the case of rectangular domains, we provide explicit values for the constants involved in the above mentioned spectral condition. Moreover, as far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

In order to precisely state our results, we need further notation. Let X be a Hilbert space, called state space, which will be identified with its dual. The inner product in X is denoted simply by •, • and the subordinated norm by • . Let A : D(A) → X be a a self-adjoint operator (possibly unbounded) on X such that Az, z 0 (z ∈ D(A)).

Such an operator will be briefly called a negative operator. We also assume that A is diagonalizable with an orthonormal basis of eigenvectors {ϕ k } k 0 and corresponding family of eigenvalues {-λ k } k 0 , where the sequence (λ k ) is positive, non decreasing and satisfies λ k → ∞. According to classical results, this holds, in particular, if A has compact resolvents. With the above assumptions on A, we have

(1.2) Az = - k 0 λ k z, ϕ k ϕ k (z ∈ D(A)), so A generates a contraction semigroup T on X satisfying (1.3) T t z = k 0 e -λ k t z, ϕ k ϕ k (t 0, z ∈ X).
Moreover, the sets (1.4)

X β := z ∈ X : k 0 1 + λ 2 k β | z, ϕ k | 2 < ∞ (β > 0)
endowed with the inner product

(1.5) y, z β = k 0 1 + λ 2 k β y, ϕ k z, ϕ k (z, y ∈ X β )
are Hilbert spaces. The scale {X β } β 0 of Hilbert spaces can be extended to a scale {X β } β∈R by defining, for every β < 0, X β as the completion of X with respect to the norm associated to the inner product (1.5). Alternatively, X -β may be defined, for β > 0, as the dual of X β with respect to the pivot space X. For every β > 0, formulas (1.2) and (1.3), with •, • standing this time for the duality between X -β and X β , provide canonical extensions for the operator A and the semigroup T to, respectively, a negative operator and a contraction semigroup on X -β -indeed, the corresponding series converge in X -β . These extensions will be still denoted by A and T. Note that, for every β ∈ R, the family {(1

+ λ 2 k ) β/2 ϕ k } k 0 is an orthonormal basis in X β .
Remark 1.1. The operator A is not explicitly necessary to construct the Hilbert spaces scale {X β } β∈R . Indeed, one only needs to consider, formally, the Hilbert space X, a non decreasing, non negative sequence {λ n } n 0 , with λ n → ∞, and an orthonormal basis {ϕ n } n 0 in X.

Let U be another Hilbert space, called input space, also identified with its dual. Let β 0 and let B ∈ L (U, X -β ) be an input operator. With the above notation, the solution w = w z of (1.1) is defined by

(1.6) w z (t) = T t z + t 0 T t-s Bu(s) ds t 0, z ∈ X, u ∈ L 2 ([0, ∞[, U ) . Note that, in general, w z ∈ C([0, ∞[, X), so that (1.6) should be understood in C([0, ∞[, X -β ).
The system (1.1) is said to be null-controllable in time T > 0 if, for every z ∈ X, there exists u ∈ L 2 ([0, T ], U ) such that w z (T ) = 0, i.e., if, for every z ∈ X, the set C T,z := u ∈ L 2 ([0, T ], U ) : w z (T ) = 0 is non empty. The quantity

C T := sup z =1 inf u∈C T,z u L 2 ([0,T ],U )
is then called the control cost. Note that, since X, U are identified with their duals, we have B * ∈ L (X β , U ). Below and in the sequel, we freely use Vinogradov's -notation. Thus a formula of the type f (x) g(x) (x ∈ X) indicates that, for all x in the set X, the inequality |f (x)| ≤ C|g(x)| holds with a suitable constant C > 0, which may depend on certain implicit parameters. In this last case, the dependence may, or may not, be indicated by annotating the symbol with appropriate subscripts.

We now state our first main result.

Theorem 1.2. Let A : D(A) → X be a negative operator and let B ∈ L (U, X -β ) for some β 0. Assume that A is diagonalizable, that {ϕ k } ∞ k=0 is

an orthonormal basis of eigenvectors with corresponding non increasing sequence of eigenvalues

{-λ k } ∞ k=0 such that lim λ k = ∞. Assume furthermore that there exists γ ∈]0, 1[ such that, for suitable positive constants d 0 , d 1 , we have (1.7) λ γ k µ |a k | 2 1/2 d 0 e d 1 µ λ γ k µ a k B * ϕ k U {a k } k 0 ∈ 2 (C), µ > 1 . Then the system (1.1) is null-controllable in any time T > 0. Moreover, given c > h gh g -g 2 d h 1 , where g := γ/(1 -γ), h := g + 1 = 1/(1 -γ), the control cost satisfies (1.8) C T T -1/2 e c/T g (T > 0),
where the implicit constant depends only on d 0 , d 1 , c, β, γ and B L (U,X -β ) .

Remark 1.3. For small T , the condition on c in the above theorem coincides with the one obtained in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF], where a larger family of generators A is considered-in particular A is not supposed to be negative or with discrete spectrum, so that assumption (1.7) is stated in a more general form. Unlike in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF], we do not assume in Theorem 1.2 that B is an admissible control operator for the semigroup generated by A-see, for instance, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Section 4.2] for the definition of this admissibility concept.

Remark 1.4. From (1.8), it follows that there exists a constant K > 0 such that

C T K/ √ T for large T . This decay rate is sharp if λ 0 = 0 since, if u 0 is a control steering the initial state ϕ 0 to zero in time T , then T 0 u 0 (t), ϕ 0 dt = -1, so that the Cauchy-Schwarz inequality clearly yields u 0 L 2 ([0,T ],U )
1/ √ T . However, if λ 0 > 0, then C T decays exponentially as T → ∞. Indeed, we can then select u := 0 on [0, T /2] and then steer z to 0 at time T by applying, on [T /2, T ], a control uniformly bounded for T > 0: that this is possible follows from Theorem 1.2.

A typical range of application for Theorem 1.2 is provided by diffusion equations (possibly fractional) in an open bounded set Ω ⊂ R n , with control acting in an arbitrary open set O ⊂ Ω. The major difficulty is then to prove (1.7). This problem is solved in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] when A := -A η 0 , η > 1 2 , γ := 1/(2η), and -A 0 is the Dirichlet Laplacian on Ω, provided that ∂Ω is of class C ∞ . The corresponding proof uses deep Carleman estimates and interpolation inequalities. In Section 3, we tackle the case of rectangular Ω by a different and fully elementary method. More precisely, we prove the following result. In particular, if Ω = [0, π] m , then, given any ε > 0, we may select (1.9)

d 1 := 3(1 + ε)m log(4πe) -3(1 + ε) log |O * |,
where O * is any hyper-rectangle included in O, and

d 0 := 2 -m/4 π m/2 1 eε log(4e) 3m/4
. By combining Theorem 1.5 (for a rectangular Ω) or the results in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] (for the case in which ∂Ω is C ∞ ) with Theorem 1.2 with γ := 1/(2η), we obtain the following statement.

Corollary 1.6. Let m ∈ N * , let Ω ⊂ R m be a rectangular domain or a bounded domain with C ∞ boundary and assume the spaces X, U and the operators A, B verify the hypotheses of Theorem 1.5. Then the system (1.1) is null-controllable in any time T > 0. Moreover, there exists a constant K > 0 such that the control cost satisfies

(1.10) C T T -1/2 exp KT -1/(2η-1) (T > 0),
where the implicit constant depends only on Ω, O and K. Moreover, in the case Ω = [0, π] m , the constant K can be chosen to be any number satisfying the condition

K > (2η) 2η/(2η-1) 2 (2η -1) -1/(2η-1) d 2η/(2η-1) 1
, where d 1 is the constant defined in (1.9).

Study of a moment problem

This section is devoted to set up the main argument of the proof of Theorem 1.2. Given T > 0, a sequence {ψ k } k 0 in a Hilbert space U and a non decreasing real sequence {λ k } k 0 , we are interested in the solvability of the moment problem which consists in finding, for every sequence

{a k } k 0 , a function Θ ∈ L 2 ([0, T ], U ) such that a n e -λnT + T 0 e -λnt Θ(t), ψ n U dt = 0 (n 0).
Appealing to a simplification of the methodology proposed in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and developed in [START_REF] Miller | On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF], [START_REF] Seidman | How violent are fast controls. III[END_REF] and [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] we give below sufficient conditions for the solvability of this moment problem and provide estimates for the solution Θ. We start by recalling a classical result from Hilbert space theory. For the sake of completeness, we provide a short proof, inspired from [11, Section 12.1].

Lemma 2.1. Suppose that W , Y , Z are Hilbert spaces, F ∈ L (W, Z) and that G ∈ L (Y, Z). Then the following statements are equivalent: (i) There exists c > 0 such that

F * z W c G * z Y (z ∈ Z); (ii) There exists H ∈ L (W, Y ) such that G H = F and H L (W,Y ) c.
Proof. That (ii) implies (i) readily follows from the facts that H * G * = F * and

H * L (Y,W ) = H L (W,Y )
. It remains to show that (i) implies (ii). Assume (i) holds. We may define a mapping

K : Ran G * → Ran F * by the formula K (G * z) = F * z for all z ∈ Z. Indeed, K is well defined since G * z 1 = G * z 2 implies F * (z 1 -z 2 ) W ≤ c G * (z 1 -z 2 ) Y = 0,
and so F * z 1 = F * z 2 . Moreover, condition (i) can be rewritten as

K (G * z) W c G * z Y (z ∈ Z).
Hence, K has a unique continuous extension to the closure Ran G * such that (2.1)

K y W c y Y (y ∈ Ran G * ).
Next, we extend K to Y by setting K y = 0 for y ∈ (Ran G * ) ⊥ . We still have

K G * z = F * z for every z ∈ Z and if y = y 1 + y 2 ∈ Y with y 1 ∈ Ran G * and y 2 ∈ (Ran G * ) ⊥ , we infer from (2.1) that K y W = K y 1 W c y 1 Y c y Y (y ∈ Y ), so that K L (Y,W ) c.
Consequently, the operator H = K * satisfies the required conditions.

The main result of this section is Proposition 2.3 below. This statement rests upon a double set of hypotheses concerning the sequence {ψ n } ∞ n=0 of vectors of U and the non decreasing sequence of non-negative real numbers {λ n } ∞ n=0 , namely (H1) (i) λ n Cn α for all n ∈ N and suitable constants α > 0, C > 0,

(ii) ψ n U M (1 + λ n ) β (n 0) for some M > 0, β > 0, and (H2)
     (i) lim n→∞ λ n = ∞,
(ii) ψ n = B * ϕ n (n 0) where {ϕ n } ∞ n=0 is an orthonormal basis of X and B ∈ L (U, X -β ) for some β > 0, where the scale of Hilbert spaces {X β } β∈R is formally constructed from the sequences (λ n ) and (ϕ n ) as described in Remark 1.1.

We exploit these assumptions in the following way (recall from the previous section the notation for the Vinogradov symbol).

Lemma 2.2. Let {ψ n } ∞ n=0 be a sequence of vectors of U and {λ n } ∞ n=0 be a non decreasing sequence of non-negative real numbers such that one of the hypotheses (H1) or (H2) is satisfied. Let {a n } ∞ n=0 ∈ 2 (C), λ > 0 and denote

F (t) := n 0 |a n | 2 e -2λnt , F λ (t) := λn λ |a n | 2 e -2λnt , F ⊥ λ (t) := F (t) -F λ (t), G λ (t) := λn λ a n e -λnt ψ n 2 U , G ⊥ λ (t) := λn>λ a n e -λnt ψ n 2 U .
Then, for all positive real numbers ε, g, ϑ and η such that ϑ + η + ε = 1, we have

(2.2) F ⊥ λ (t) + G ⊥ λ (t) e -2ϑλt+ε/t g F (ηt) (t > 0).
The implicit constant may depend on α, β, C, ε, g, M in the first case, and on B, β, ε, g in the second.

Proof. It clearly suffices to prove the upper bound for G ⊥ λ . Assume first that (H1) holds. Then, by the Cauchy-Schwarz inequality, (2.3)

G ⊥ λ (t) M 2 λn>λ e -3ελnt/2 λn>λ |a n | 2 e -(2-3ε/2)λnt (1 + λ n ) 2β β,ε M 2 (1 + t -2β ) n 0 e -3Cεn α t/2 λn>λ |a n | 2 e -(2-2ε)λnt .
Since

n 0 e -3Cεn α /2 1 + ∞ 0 e -3Cεx α /2 dx = 1 + Γ(1 + 1/α) (3Cεt/2) 1/α ,
we obtain

G ⊥ λ (t) (1 + t -1/α )(1 + t -2β )e -2ϑλt F (ηt) e -2ϑλt+ε/t g F (ηt).
Assume now that (H2) holds. Then, Parseval's formula implies that

G ⊥ λ (t) = B * λn>λ a n e -λnt ϕ n 2 U B 2 L (U,X -β ) λn>λ |a n | 2 e -2λnt (1 + λ 2 n ) β B 2 L (U,X -β ) (1 + t -2β )e -2λϑt F (ηt),
where, in the last stage, we used the bound

(1 + λ 2 n ) β e -2λn(ϑ+η+ε)t (1 + t -2β )e -2ϑλt-2ηλnt (λ n > λ, t > 0).
This readily implies the required conclusion.

We are now in a position to state the main result in this section.

Proposition 2.3. With the notation and assumptions of Lemma 2.2, put

G(t) := n 0 a n e -λnt ψ n 2 U (t > 0),
and assume furthermore that there exists γ ∈]0, 1[ such that, for suitable positive constants d 0 , d 1 , we have

(2.4) λ γ k µ |a k | 2 1/2 d 0 e d 1 µ λ γ k µ a k ψ k U {a k } k 0 ∈ 2 (C), µ > 1 . Let g := γ/(1 -γ), h := 1/(1 -γ). Given D > 2 h gh g -g 2 d h 1 , we have (2.5) F (T ) e D/T g T T 0 G(t) dt (T > 0).
The implicit constant above may depend upon γ, d 0 , d 1 , D, and on α, β, C, M if hypothesis (H1) holds, or B, β if assumption (H2) is satisfied.

On the null-controllability of diffusion equations

Proof. Let r > 1 and {a k } k 0 ∈ 2 (C). By (2.4), we have

F λ (t) d 2 0 e 2d 1 λ γ G λ (t) λ > 0, T /r t T .
Integrating this with respect to t and using the trivial bound

(r -1)T r e -2λ k T T T /r e -2λ k t dt (k ∈ N),
we obtain

(2.6) F λ (T ) K 1 T e K 2 λ γ T T /r G λ (t) dt (T > 0), with K 1 := rd 2 0 /(r -1), K 2 := 2d 1 .
Let ε > 0 and let ϑ, σ > 0 be such that ε = 1ϑrσ. Recall that g := γ/(1γ) and h := 1/(1γ). Using Lemma 2.2 and (2.6), it is not difficult to check that (2.7)

F (T ) e K 2 λ γ +ε/T g T T T /r G(t) dt + e K 2 λ γ -2ϑλT +ε/T g F (σT ),
where the implicit constant may depend upon K 1 and all parameters indicated in the statement of Lemma 2.2. Given ν > 0, we select λ := {K 2 ν/ϑσT } h , so that

K 2 λ γ -2ϑλT = K h 2 ν g (ϑσT ) g 1 - 2ν σ = - K h 2 ν g (2ν -σ) ϑ g σ h T g • Writing D 0 := K h 2 ν g ϑ g σ g , E 0 := K h 2 ν g (2ν -σ) ϑ g σ h , equation (2.7) becomes (2.8) F (T ) K T e D/T g T T /r G(t) dt + Ke -E/T g F (σT ),
where K is a suitable constant and

(2.9)

D := D 0 + ε, E := E 0 -ε.
Selecting ν so large that

ν > 1 2σ g-1 + εσϑ g 2K h 2 ν g , we have (2.10) E > D(1/σ g -1)
.

Applying (2.8) to F (σT ) instead of F (T ), we thus arrive at

F (T ) K T e D/T g T T /r G(t) dt + K 2 σT e D 1 /T g σT σT /r G(t) dt + K 2 e -E 1 /T g F (σ 2 T )
where

D 1 := D/σ g -E, E 1 := E(1 + σ g )/σ g .
After N iterations, we thus get (2.11)

F (T ) K T e D/T g T T /r G(t) dt + 1 j N K j+1 σ j T e D j /T g σ j T σ j T /r G(t) dt +K N +1 e -E N /T g F σ N +1 T ,
with

D j := D -E(σ g + • • • + σ gj ) σ gj = 1 σ gj D - E(1 -σ jg ) 1/σ g -1 , E N := E(1 + • • • + σ gN ) σ gN •
Observe that sup t 0 F (t) a 2 (C) , so that the last term in (2.11) tends to 0 as N → ∞. Moreover, we infer from (2.10) that D j < D for all j and that D j tends exponentially to -∞ as j → ∞. Therefore, under the above restrictions on the various parameters, we obtain

j 1 K j+1 σ j T e D j /T g e D/T g T if, say, T 1.
The same result holds for T > 1 provided we increase, if needed, the value of ν to ensure that D 1 = -LT g where L is a constant possibly depending on all parameters except T and such that L > 2K/σ: then, we have e D/T g 1 and the alteration of D is actually irrelevant.

Letting N → ∞ in (2.11), we finally obtain

F (T ) e D/T g T T 0 G(t) dt,
where the implicit constant is restricted as in the statement. This plainly implies the stated result when T > 1 since, as noted above, the exponential factor is bounded with our choice for ν. When T 1, we may select any ϑ < 1/h, σ < g/h, ν > 1 2 σ 1-g provided r is sufficiently close to 1 and ε is small enough. Then, D can be taken as any number strictly exceeding 2 -g h gh g -g 2 K h 2 . Since K 2 = 2d 1 , we indeed get that (2.5) holds whenever D > 2 h gh g -g 2 d h 1 .

Proposition 2.4. With the notation and assumptions of Proposition 2.3, for every T > 0 there exists a bounded operator

H T ∈ L ( 2 (C), L 2 ([0, T ], U )) such that, for every {a n } n 0 ∈ 2 (C), we have (2.12) a n e -λnT + T 0 e -λnt H T a(t), ψ n U dt = 0 (n 0).
Moreover, for every c > h gh g -g 2 d h 1 , we have

(2.13) H T e c/T g √ T (T > 0).
The implicit constant above may depend upon γ, d 0 , d 1 , D and on α, β, C, M if hypothesis (H1) holds, or B, β if assumption (H2) is satisfied.

Proof. Write Y := L 2 ([0, T ], U ), Z := 2 (C).
Here Y and Z are endowed with the canonical scalar products

u, v Y := T 0 u(t), v(t) U dt, a, b Z := n 0 a n b n . Given ε > 0, we consider G ε ∈ L (Y, Z) defined by (2.14) (G ε u) n := u, e -λn(t+ε) ψ n Y (n 0).
Then for any a ∈ Z, we have

G ε u, a Z = n 0 a n u, e -λn(t+ε) ψ n Y = u, n 0 a n e -λn(t+ε) ψ n Y .
Therefore,

(2.15)

G * ε a = n 0
a n e -λn(t+ε) ψ n (a ∈ 2 (C)).

Consider also the self-adjoint operator F ε ∈ L (Z) defined by

(F ε a) n = -e -λn(T +ε) a n (a ∈ Z, n 0).
Applying Proposition 2.3 to the sequence {a n e -ελn } ∞ n=0 , we obtain that the bound

F * ε a Z e c/T g √ T G * ε a Y ,
holds for every c > h gh g -g 2 d h 1 . Note that the implicit constant above does not depend on ε. By Lemma 2.1 with W = Z, it follows that there exists

H ε ∈ L (Z, Y ) such that G ε H ε = F ε . Moreover, H ε e c/T g T -1/2
where the implied constant is independent of ε > 0. Thus, for any a ∈ Z, we have

(F ε a) n = -a n e -λn(T +ε) = (G ε H ε a(t)) n = T 0 H ε a(t), e -λn(t+ε) ψ n U dt (n 0).
Dividing through by e -λnε , we see that H T := H ε satisfies (2.12) and (2.13).

Remark 2.5. Albeit our proof of Theorem 1.2 (see Section 3) only appeals to assumption (H2), we stated Proposition 2.4 in a more general form: indeed, its validity under assumption (H1) is of independent interest. This latter hypothesis can also be used for systems governed by diffusion equations (for instance in the proof of Corollary 1.6) since condition (i) in (H1) is satisfied when -A 0 is the Dirichlet Laplacian; this follows from Weyl's formula-in fact a simpler classical result (see, for instance, [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Proposition 3.6.9]) suffices.

Proof of the main results

We first show that Theorem 1.2 is a simple consequence of Proposition 2.4.

Proof of Theorem 1.2. Set a n := z, ϕ n (n 0). From (1.3) and (1.6), we see that the null-controllability of (1.1) in time T is equivalent to the existence, for every z ∈ X, of a function u ∈ L 2 ([0, T ], U ) such that a n e -λnT + T 0 e -λn(T -t) u(t), B * ϕ n U dt = 0 (n 0). Now Proposition 2.4 implies that the function

u(t) := (H T a)(T -t) (t ∈ [0, T ]),
satisfies the above condition. Consequently, (1.1) is null-controllable in any time T > 0. Finally, the cost estimate (1.8) readily follows from (2.13).

Our proof of Theorem 1.5 is based on the following result of Turán [START_REF] Turán | On a theorem of Littlewood[END_REF]-see also Lemma 1 in Montgomery [9, p.89

]. Lemma 3.1. Let N ∈ N * , Λ := [-N, N ] ∩ Z and let {a n } n∈Λ be a complex sequence. Put f (x) := n∈Λ a n e inx (x ∈ R).
Then, for every subinterval I of [-π, π] with length |I| = 4πeL, we have

sup x∈I |f (x)| L 2N max x∈[-π,π] |f (x)|.
This result opens the way to a comparison statement between the L 2 -norms of a trigonometric polynomial over the full torus and over a restricted interval. Proposition 3.2. With the notation in Lemma 3.1, we have

(3.1) I |f (x)| 2 dx L 6N 8 √ 2N 3/2 n∈Λ |a n | 2 . Proof. Let x 0 ∈ I be such that |f (x 0 )| = f L ∞ (I)
. By the Cauchy-Schwarz inequality, we have

(3.2) |f (x)| N 3/2 2 n∈Λ |a n | 2 √ 2N 3/2 f L ∞ [-π,π] (x ∈ R). Set (3.3) h := L 2N √ 8N 3/2 , so that it follows from (3.2) that h|f (x)| 1 2 f L ∞ [-π,π] L 2N (x ∈ R),
and in turn, from Lemma 3.1,

(3.4) h |f (x 0 )| 2 f L ∞ [-π,π] •
The above inequality implies that

min |x-x 0 | h |f (x)| 1 2 |f (x 0 )|.
Moreover, since h < 2πeL, at least one of the intervals [x 0h, x 0 ] and [x 0 , x 0 + h] is contained in I. Therefore

I |f (x)| 2 dx 1 4 h|f (x 0 )| 2 .
Appealing to Lemma 3.1 again and using (3.3), it follows that

I |f (x)| 2 dx L 6N 4 √ 8N 3/2 f 2 L ∞ [-π,π] L 6N 4 √ 8N 3/2 n∈Λ |a n | 2 .
We now derive a straightforward corollary of the above proposition, adapted to sine or cosine series in several variables. More precisely, for every m 1, we define

(3.5) ϕ α,m (x) = 2 π m/2 1 j m sin (α j x j ) α ∈ N m , x ∈ [0, π] m , and (3.6
) Φα, m(x) = 2 π m/2 1 j m cos (α j x j ) α ∈ N m , x ∈ [0, π] m . Corollary 3.3. Let m, N 1, D ⊂ [0, N ] m ∩ N m , and P (x) := α∈D b α α,m (x) (x ∈ R m )
be a trigonometrical polynomial with complex coefficients, where either α,m := ϕ α,m for all α or α,m := Φ α,m for all α. Moreover, assume in the former case that b α := 0 whenever 1 j m α j = 0.

If R is a hyper-rectangle included in [0, π] m with measure (4πe) m ϑ > 0, we have

(3.7) α∈D |b α | 2 (4π) m 2N 3 m/2 ϑ -6N R |P (x)| 2 dx.
Proof. We only consider the case ( α,m ) = (ϕ α,m ), since the other case is similar. We argue by induction upon m. When m = 1, we have (3m/4eλ) 3m/4 e λx , which is valid for all λ > 0, m 1, x 0. Since B * ϕ = ϕ| O for every ϕ ∈ L 2 (Ω), inequality (3.9) is precisely (1.7) with γ := 1/(2η) ∈]0, 1[. We proceed similarly when -A 0 is the the Neumann Laplacian. The family {Φ α,m : α ∈ N m } defined in (3.6) is an orthonormal basis of L 2 (Ω) formed of eigenvectors of -A η 0 on Ω and corresponding to the sequence of eigenvalues (λ α ) still given by (3.8), but with now α ∈ N m . From this point on, the argument is unchanged.

Theorem 1 . 5 .

 15 Let m ∈ N * , Ω ⊂ R m be a rectangular domain, X := L 2 (Ω), let A := -A η 0 , where -A 0 is the Dirichlet or the Neumann Laplacian in Ω and η > 1/2. Set U := L 2 (O) and define B ∈ L (U, X) by the formula Bv := v1l O (v ∈ U ), where 1l O is the characteristic function of a non empty open subset O of Ω. Then assumption (1.7) holds with γ := 1/(2η).

2 1 n N |b n | 2 .µ > 1 . 4 = 2 4 ,

 221424 i √ 2πP (x) = n∈Λ a n e inx , with Λ := [-N, N ] ∩ Z and a n = sgn(n)b n (n ∈ Λ). By Proposition 3.2, we thus obtain 8π √ 2N 3/2 ϑ -6N R |P (x)| 2 dx n∈Λ |a n | 2 = This is the required inequality for m = 1. Let m 2 and assume (3.7) holds for m -1. We may write R = S × I with S ⊂ [0, π] m-1 , |S | = 4πeτ , |I| = 4πeσ, and τ σ = ϑ. Moreover, settingD n := {β ∈ (N * ) m-1 : (β, n) ∈ D} (1 n N ), P n (x) := β∈Dn b βn ϕ β,m-1 (x) 1 n N, x ∈ R m-1 ,we have for all x ∈ R m-1 , in view of (3.1),I |P (x, x m )| 2 dx m = I 1 n N P n (x)ϕ n,1 (x m ) 2 dx m (σ/e) 6N 4π √ 2N 3/2 |n| N |P n (x)| 2 ,hence, by the induction hypothesis,R |P (x)| 2 dx = S I |P (x, x m )| 2 dx m dx σ 6N 4π √ 2N 3/2 |n| N S |P n (x)| 2 dx (τ σ) 6N (4π) m (2N 3 ) m/2 |n| N β∈Dn |b βn | 2 .Proof of Theorem 1.5. For the sake of simplicity, we select Ω =]0, π[ m . Let -A 0 be the Dirichlet Laplacian on Ω. Then, the family {ϕ α,m : α ∈ (N * ) m } defined in (3.5) is an orthonormal basis of L 2 (Ω) formed of eigenvectors of -A η 0 on Ω and corresponding to the sequence of eigenvalues (λ α ) given by (3.8) λ α := α 2η = N * ) m . Corollary 3.3 immediately furnishes constants d 0 , d 1 such that, for every {a α } ∈ 2 ((N * ) n , C), Indeed, given any ε > 0, we may select d 1 := 3(1 + ε)m log(4πe) -3(1 + ε) log |O * |, for any hyper-rectangle O * included in O and then, since d 1 3m log(4e), d 0 := 2 5m/4 π m/2 1 4eε log(4e) 3m/-m/4 π m/2 1 eε log(4e) 3m/in view of the elementary inequality x 3m/4

Acknowledgements. The author are grateful to Luc Miller for drawing their attention on his submitted work [8] and for helpful discussions.


Gérald Tenenbaum

Alekseevich Karatsuba

A note on the distribution of some additive functions

Keywords: probabilistic number theory, Erdős-Wintner theorem, arithmetic functions, additive functions, concentration functions, distribution of arithmetic functions, modulus of continuity

Let f denote an additive arithmetical function with continuous limiting distribution F on the integers. Then f also has a limiting distribution G on shifted primes. Under some growth conditions on the values of f at primes, we provide optimal lower bounds for the modulus of continuity of F and G, at all points from a specified infinite set.

Introduction

The famous Erdős-Wintner theorem [START_REF] Erdős | Additive arithmetical functions and statistical independence[END_REF] states that a real, additive arithmetical function f has a limiting distribution if, and only if, the following three series converge [START_REF] Babu | Absolutely continuous distribution functions of additive functions f (p) = (log p) -a , a > 0[END_REF]

|f (p)| 1 ϕ(p) p , |f (p)| 1 f (p) 2 p , |f (p)|>1 1 p ,
where, here and in the sequel, the letter p generically denotes a prime number. Moreover, the characteristic function of the limit law F (z) is the given by the formula

(2)

F (τ ) := R e iτ z dF (z) = p 1 - 1 p ν 0 e iτ f (p ν ) p ν .
It has been long conjectured that the same conditions are also necessary and sufficient for the existence of a limiting distribution on a sequence of shifted primes, i.e. for the weak convergence, as x → ∞, of the distribution functions

1 π(x) p x f (p+a) z 1
to a distribution function G(z) whenever a is a non-zero integer. The sufficiency part of this conjecture was established by Kátai [START_REF] Kátai | On distribution of arithmetical functions on the set prime plus one[END_REF] using the Bombieri-Vinogradov theorem, whereas the more difficult necessary part, known as the Erdős-Kubilius conjecture, has been proved by Hildebrand [START_REF] Hildebrand | Additive and multiplicative functions on shifted primes[END_REF]. In this setting, the characteristic function of the limit law takes the form

(3) G(τ ) := p a 1 - 1 p -1 + ν 1 e iτ f (p ν ) p ν
In both cases, it follows respectively from classical results of Jessen-Wintner and of Lévy that the limiting distribution is always of pure type and that it is continuous if, and only if, ( 4)

f (p) =0 1 p = ∞.
In the classical situation (2), some attention has been focused along the years on the local behaviour of the limit law. We refer the reader to the recent work [START_REF] De La Bretèche | Sur la concentration de certaines fonctions additives[END_REF] for a review of results on the concentration function of the law F , and to the papers of Toulmonde [START_REF] Toulmonde | Module de continuité de la fonction de repartition de ϕ(n)/n[END_REF], [START_REF] Toulmonde | Sur les variations de la fonction de répartition de ϕ(n)/n[END_REF], [START_REF] Toulmonde | Vincent Comportement au voisinage de 1 de la fonction de répartition de ϕ(n)/n[END_REF], and Tenenbaum-Toulmonde [START_REF] Tenenbaum | Sur le comportement local de la répartition de l'indicatrice d'Euler[END_REF] for progress in the study of the modulus of continuity in the case f (n) := log{n/ϕ(n)}, where ϕ denotes the Euler totient.

In the analogous problem for shifted primes, corresponding to formula (3), Deshouillers and Hassani [START_REF] Deshouillers | A note on the distribution function of ϕ(p-1)[END_REF] recently considered the question of differentiability of the distribution function G(z) when f (n) = log{n/ϕ(n)}, and showed that the left derivative is infinite at all points in f (2N * ).

The main purpose of this note is to supply fairly general hypotheses implying that F and G both have infinite derivatives at all points from a specified infinite set, and indeed to provide explicit lower estimates for the moduli of continuity at these points.

We write

N a := {n ∈ N * : (n, a) = 1} and put v := 1 if 2 a, v := 0 if 2 | a.
Theorem 1. Let δ > 0 and c := δ/(1 + δ). Under hypotheses (4) and

(5)

f (p) 1/(log p) 1+δ ,
we have, as ε approaches 0,

F (z + ε) -F (z -ε) z ε 1-c (z ∈ f (N * )), G(z + ε) -G(z -ε) z ε 1-c (z ∈ f (2 v N a )).
It is interesting to note that, in the example of a strongly additive function f satisfying f (p) := 1/(log p) 1+δ with 0 < δ < 1, we know from a result of Babu (see also Exercise 259 of [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF], with solution in [START_REF] Tenenbaum ; With | Exercices corrigés de théorie analytique et probabiliste des nombres, Cours spécialisés[END_REF]) that F is absolutely continuous. This cannot happen when f (p) 1/p δ , since a result of Erdős [START_REF] Erdős | On the smoothness of the asymptotic distribution of additive arithmetical functions[END_REF] then guarantees that F is purely singular. Such property also holds for G, as can be shown by adapting the solution, given in [START_REF] Tenenbaum ; With | Exercices corrigés de théorie analytique et probabiliste des nombres, Cours spécialisés[END_REF], of Exercise 256 of [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF].

We also remark that our result above provides, as a weak consequence, the estimates

Q F (ε) ε 1-c , Q G (ε) ε 1-c (0 < ε < 1)
for the concentration functions associated to F and G respectively. While the second appears to be new, the first of these lower bounds recovers that of Erdős and Kátai [START_REF] Erdos | On the concentration of distribution of additive functions[END_REF], stated in the case of strongly additive functions such that f (p) = 1/(log p) 1+δ and which is optimal up to a factor (log log 1/ε) 2 -see [START_REF] De La Bretèche | Sur la concentration de certaines fonctions additives[END_REF] for an improvement.

The author takes pleasure in expressing here warm thanks to Jean-Marc Deshouillers, for drawing his attention to this topic and fruitful subsequent exchanges, and to Régis de la Bretèche, for helpful discussions on this matter.

Proof of Theorem 1

We restrict to the case of G with a = ±1, since the cases of arbitrary a and of F are analogous. We may plainly assume δ < 1.

Given y > 2, we let

G y (τ ) := 2<p y 1 - 1 p -1 ν 1 e iτ f (2 ν ) 2 ν 2<p y 1 + p -1 p -2 ν 1 e iτ f (p ν ) p ν
denote the characteristic function of the distribution function

(6) G y (z) := 2<p y 1 - 1 p -1 P + (n) y f (2n) z 1 k(2n)
where k is the multiplicative function defined by k(2 ν ) := 2 ν (ν 1), k(p ν ) := (p -2)p ν /(p -1) (ν 1, p > 2). We have

G y (τ ) = G(τ ) 1 + O τ /(log y) 1+δ |τ | (log y) 1+δ .
Thus the Berry-Esseen inequality implies, for all T 1,

sup z∈R |G(z) -G y (z)| Q G 1 T + T (log y) 1+δ ,
where Q G designates the concentration function associated to G. Selecting T = T y := (log y) (1+δ)/2 , we obtain [START_REF] Hildebrand | Additive and multiplicative functions on shifted primes[END_REF] sup

z∈R |G(z) -G y (z)| Q G 1 T y .
Now, for all ε > 0, 2 < v y, and letting r denote an integer all of whose prime factors belong to ]v, y], we have

|f (r)|>ε 1 k(r) r 1 k(r) e -(log v) 1+δ {ε-|f (r)|} e -ε(log v) 1+δ log y log v •
For fixed z := f (2m), sufficiently small η > 0, ε > 0 and large y, we select v := exp{(ηε) c-1 } and thus obtain, with a suitable constant K,

P + (n) y |f (2n)-z| ε 1 k(2n) 1 k(2m) |f (r)| ε 1 k(r) 1 k(m) v<p y 1 + 1 p -2 -Ke -1/η log y log v η ε 1-c log y k(m) •
Inserting back in ( 6) taking [START_REF] Hildebrand | Additive and multiplicative functions on shifted primes[END_REF] into account, we get, with suitable c 0 > 0, c 1 > 0,

G(z + ε) -G(z -ε) c 0 ε 1-c k(m) -c 1 Q G 1 T y
whence, letting y tend to infinity,

G(z + ε) -G(z -ε) ε 1-c k(m) •


Mikhail Isaev
email: mikhail.isaev@anu.edu.au

Brendan D Mckay

On a bound of Hoeffding in the complex case

It was proved by Hoeffding in 1963 that a real random variable X confined to [a, b] satisfies E e X-E X ≤ e (b-a) 2 /8 . We generalise this to complex random variables.

Introduction

A celebrated concentration inequality of Hoeffding relies on the following bound.

Lemma 1. 1 ([4]). Let X be a real random variable such that a ≤ X ≤ b. Then

1 ≤ E e X-E X ≤ e (b-a) 2 /8 .
Given the hundreds of references to this inequality in the literature, we believe that a similarly tight bound for complex random variables may also have application. Indeed, our ongoing investigation of complex martingales related to multidimensional asymptotics encounters such a need. Our aim, therefore, is to find a complex analogue of Lemma 1.1.

There are several possible complex replacements for the real bounds a ≤ X ≤ b. A natural choice would be to confine Z to a disk of given radius, but we will use the weaker condition that the support of Z has bounded diameter. This measure of spread naturally arises in the study of separately Lipschitz functions [START_REF] Fan | Exponential inequalities for martingales with applications[END_REF], also called functions satisfying the bounded difference condition [START_REF] Boucheron | Concentration Inequalities: A Nonasymptotic Theory of Independence[END_REF]. For a complex random variable Z, define the diameter of Z to be

diam Z = inf c ∈ R | P (|Z 1 -Z 2 | > c) = 0 , (1.1)
where Z 1 , Z 2 are independent copies of Z, with the infimum of an empty set taken to be ∞. Since Hoeffding's bound states that E e X-E X is concentrated near 1, the natural complex analogue is to bound the distance of E e Z-E Z from 1. This is the nature of our main theorem.

Theorem 1.1. Let Z be a complex random variable with diam Z ≤ d. Then

|E e Z-E Z -1| ≤ e d 2 /8 -1.
Since |Z -E Z| ≤ α implies diam Z ≤ 2α, Theorem 1.1 has a simple consequence, which is a complex version of an inequality used many times in proving Azuma-type inequalities.

Corollary 1.1. Let Z be a complex random variable with |Z -E Z| ≤ α a.s.. Then

|E e Z-E Z -1| ≤ e α 2 /2 -1.
Note that in both the theorem and its corollary, distrubutions supported on {-1, +1} are enough to show that the constants (respectively 1 8 and 1 2 ) cannot be reduced. Corollary 1.1 and weaker versions of Theorem 1.1 can be proved by simpler means. For example, the referee noted that Corollary 1.1 (which implies Theorem 1.1 with constant 1 2 instead of 1 8 ) can be proved by representing complex numbers by real matrices and applying result of Tropp [START_REF] Tropp | User friendly tail bounds for sums of random matrices[END_REF].

Hoeffding actually found the best possible bound on e X-E X in the real case, as we recall in the next section. In Section 2.1 we show that for d ≤ 3.12 the same tighter bound holds in the complex case too, making use of a lemma that only random variables with support of at most three points need to be considered. Then in Section 2.2 we complete the proof of Theorem 1.1 for all d.

Results

Hoeffding's paper [START_REF] Hoeffding | Probability Inequalities for sums of bounded random variables[END_REF] used the convexity of the exponential function to find the tightest possible bound in the real case. For d > 0, consider the random variable X d supported on {0, d} with

P (X d = d) = e d -1 -d d(e d -1) = 1 2 -1 12d + O(d 3 ). We find that E e X d -E X d -1 = G(d),
where

G(d) = exp -e d -1-d e d -1 -2 exp de d -e d +1 e d -1 + exp 2de d -e d -d+1 e d -1 d(e d -1) -1 = 1 8 d 2 + 7 1152 d 4 + O(d 6 ) = exp 1 8 d 2 -1 576 d 4 + O(d 6 ) -1.
We can now state Hoeffding's bound in its strongest form.

Lemma 2.1. Let X be a real random variable such that a ≤ X ≤ b.

Then |E e X-E X -1| ≤ G(b -a) ≤ e (b-a) 2 /8 -1,
where the first inequality holds with equality if and only if X = X b-a + a almost surely.

The complex case: tight bound for small diameter

Let Z 1 , . . . , Z n be complex random variables and let c 1 , . . . , c n be nonnegative real numbers with

c 1 + • • • + c n = 1. Define the mixture Z = Mix c 1 ,...,cn (Z 1 , . . . , Z n ) by P (Z ∈ A) = n k=1 c k P (Z k ∈ A)
for every measurable set A ⊆ C. A standard property of mixtures is that

E F Mix c 1 ,...,cn (Z 1 , . . . , Z n ) = n j=1 c j E F (Z j ) (2.1)
for any measurable function F : C → C for which the expectations exist. Let Z d be the class of all complex random variables Z with E Z = 0 and diam Z ≤ d, and let Z (k) d be the subclass of Z d consisting of those variables supported on at most k points.

Lemma 2.2. Let F : {z ∈ C | |z| ≤ d} → C be continuous. Then sup Z∈Z d |E F (Z)| = sup Z∈Z (3) d |E F (Z)|.
Proof. This is an example of the "Carathéodory Principle", see for example [START_REF] Hoeffding | The extrema of the expected value of a function of independent random variables[END_REF][START_REF] Mulholland | Representation theorems for distribution functions[END_REF][START_REF] Pinelis | Tchebycheff systems and extremal problems for generalized moments: a brief survey[END_REF]. Since we didn't find a statement in the literature that exactly matches our needs, we outline the proof.

First, by a simple induction, any Z ∈ Z d with finite support can be written as a mixture of members of Z

(3) d . This is true because, for any finite set of points in C having the origin in its convex hull, there is a subset of three or fewer points having the origin in its convex hull. By (2.1), this implies that the lemma holds when Z has finite support.

For more arbitrary Z ∈ Z d , we can use the continuity of F to show that for any ε > 0 there is Z ∈ Z d of finite support such that E Z = 0 and |E F (Z) -E F (Z )| ≤ ε. Allowing ε to tend to 0 completes the proof.

We now return to Hoeffding's bound using Lemma 2.2 for F (z) = e z -1. Obviously |E e Z-E Z -1| = 0 if Z is constant, so we need to consider the cases of 2-point and 3-point supports. Since a random variable supported on 3 collinear points is a mixture of two random variables supported on 2 points, in the case of 3 points only the non-collinear case needs to be considered.

Lemma 2.3. If Z ∈ Z (2) d , then |E e Z -1| ≤ G(d).
Proof. The case of real Z is treated in Lemma 2.1. More generally, since E Z = 0, Z = e iθ X where X is real. Since E X = 0, there are x, x ≥ 0 such that X has support -x with probability x /(x + x ) and x with probability x/(x + x ). For any odd k we can calculate that

E X k = x k 1 + ρ (ρ k -ρ), where ρ = x /x.
This shows that either X or -X has only nonnegative moments. By adding π to θ if necessary, we assume that the former holds. Now, recalling that E Z = E X = 0, we can calculate

|E e Z -1| = ∞ k=2 1 k! E Z k = ∞ k=2 1 k! e ikθ E X k ≤ ∞ k=2 1 k! E X k = E e X -1, which implies that |E e Z -1| ≤ G(d) by the real case.
The case of a support of three points is considerably more difficult. To begin, define

d 0 = sup {d ∈ R | E e Z ≥ 0 for all Z ∈ Z d }.
Since e z ≥ 0 for |z| ≤ π/2 we see that d 0 ≥ π/2. The actual value is almost twice as large.

Lemma 2.4. d 0 ≈ 3.120491233.

Proof. By applying Lemma 2.2 to F (z) = e d -z, we see that only supports of two points or three non-collinear points need to be considered. For supports of two points, we have by definition

inf{ E e Z | Z ∈ Z (2) d } = inf ∈[0,d],x∈[0,1],θ∈[-π,π] xe (1-x)e iθ + (1 -x)e -xe iθ .
There appears to be no closed form for the infimum. However, careful numerical computation shows that the infimum crosses 0 at d = d 2 ≈ 3.120491233, which occurs when = d, x ≈ 0.636527202 and θ ≈ 1.9198934984.

If d 0 < d 2 , there is some d < d 0 such that 0 < inf{ E e Z | Z ∈ Z (3) d } < inf{ E e Z | Z ∈ Z (2) d }.
By the compactness of Z

d , the first infimum is realised by some Z ∈ Z

(3) d with a support of three non-collinear points {z 1 , z 2 , z 3 }. Define Z (x,y) to be the random variable with the same support, but mean at x + iy (thus Z = Z 0,0 ). As is well known, x, y determine the probabilities at z 1 , z 2 , z 3 linearly, so for some complex constants A, B, C we have

E e Z (x,y) -E Z (x,y) -1 = e -x-iy (Ax + By + C) -1 = C -1 + (A -C)x + (B -iC)y + O(x 2 + y 2 ), (2.2)
valid whenever x + iy lies in the convex hull of {z 1 , z 2 , z 3 }.

In order for Z to be a local minimum for the real part, the coefficients of x and y in (2.2) must be purely imaginary. Therefore, for some v, w ∈ R, A = C + iv and B = iC + iw. Substituting in these values and writing C = c 0 + ic 1 we find

E e Z (x,y) -(x+iy) = c 0 + [x, y]R[x, y] T + O(|x| 3 + |y| 3 ), where R = -1 2 c 0 1 2 (c 1 + v) 1 2 (c 1 + v) 1 2 c 0 + w . The smallest eigenvalue of R is 1 2 w -1 2 (w + c 0 ) 2 + (v + c 1 )
2 , which is clearly negative for c 0 > 0. Thus (x, y) = (0, 0) is not a local minimum for E e Z (x,y) -E Z (x,y) , contrary to our assumption. This proves that

d 0 = d 2 . Theorem 2.1. If diam Z = d ≤ d 0 , then |E e Z -1| ≤ G(d).
Proof. We can rely on continuity to assume that d < d 0 .

By Lemma 2.2 with F (z) = e z -1, we have sup

Z∈Z d |E e Z -1| = sup Z∈Z (3)
d |E e Z -1|, and by compactness the supremum is achieved for some Z ∈ Z

d . We first show that no such local maximum occurs when the support consists of 3 non-collinear points, implying that it occurs in Z where the support of Z consists of three non-collinear points. Let Z (x,y) be the random variable with the same support as Z, but mean at x + iy. As in the proof of Lemma 2.4, (2.2) holds. Since we are assuming this to be a local maximum for |E e Z -1|, A -C and B -iC must be orthogonal to C -1, i.e. real multiples of i(C -1). That is, for some v, w ∈ R, A = C + iv(C -1) and B = iC + iw(C -1). Substituting in these values, writing C = c 0 + ic 1 for c 0 , c 1 ∈ R, and defining ∆ = |C -1| we can expand

E e Z (x,y) -E Z (x,y) -1 2 = |C -1| 2 + [x, y]Q[x, y] T + O(|x| 3 + |y| 3 ),
where

Q = ∆v 2 + c 0 -c 2 1 -c 2 0 ∆vw -c 1 ∆vw -c 1 ∆w 2 + c 0 -1 .
We now show that Q cannot be negative semidefinite. If ∆ = 0 then E e Z -1 = 0, which is clearly not a maximum, so assume ∆ > 0. The trace of Q is ∆(v 2 + w 2 -1), which is impossible for a negative semidefinite matrix if v 2 + w 2 > 1, so assume

v 2 + w 2 ≤ 1. The determinant of Q is ∆ -w 2 c 2 0 -(1 -v 2 -w 2 )c 0 -(v -wc 1 )
2 , which is negative (since d < d 0 implies c 0 > 0), which is also impossible for a negative semidefinite matrix. Therefore, there is no local maximum here and the proof is complete.

We have no reason to believe that Theorem 2.1 requires the condition d ≤ d 0 , and expect that it is true for all d. However, the same proof is insufficient since it is possible for local maxima to occur for supports of three points. However, we can now complete the proof of Theorem 1.1 for all d.

Proof of Theorem 1.1

We need two technical bounds whose uninteresting proofs are omitted, and a standard result on planar sets. Lemma 2.5. For t ≥ 3 we have G(t) + 1 ≤ 0.9 e t 2 /8 .

(2.3)

G(2t) + 1 ≤ 1.65 e t 2 /8 (2.4)
Lemma 2.6. Let Z be a bounded complex random variable. Then Z is almost surely confined to some closed disk of radius 1 √ 3 diam Z.

Proof. This follows from a standard result on convex sets, see [5, Thm. 12.3] for example. An equilateral triangle shows that the constant cannot be reduced.

Proof of Theorem 1.1. For d ≤ 3, the theorem follows from Theorem 2.1, so we can assume that d ≥ 3. By Lemma 2.6, there is some a ∈ C such that |Z -a| ≤ 1 √ 3 d. We will find two bounds on |E Ze Z |. First we argue that

|E Ze Z | ≤ |E(Z -a)e Z | + |a E e Z | ≤ 1 √ 3 d E |e Z | + |a| |E e Z | ≤ 1 √ 3 0.9 de d 2 /8 + |a| |E e Z |, (2.5)
where we have used Lemma 2.1 to bound E|e Z | = E e Z , and (2.3). On the other hand

|E Ze Z | = |E Z(e Z -E e Z )| ≤ E|Z| 2 E|e Z -E e Z | 2 ≤ E|Z -a| 2 -|a| 2 E|e 2Z | -|E e Z | 2 ≤ d 2 /3 -|a| 2 1.65 2 e d 2 /4 -|E e Z | 2 , (2.6)
where we used Lemma 2.1 to bound E e 2 Z , and (2.4).

Taking the average of (2.5) and (2.6), we have We can now complete the proof. We have E e Z -1 = E e 3Z/d -1 +

|E Ze Z | ≤ 1

Conclusions

In conclusion, we note some questions that we have not answered. Our prediction that Theorem 2.1 holds for all d is one of them. We can also ask for an exact description of the

( 2 )

 2 d . By Lemma 2.3, this will complete the proof. So, suppose that the maximum value of |E e Z -1| for Z ∈ Z (3) d

√ 3 de d 2 1 √ 3

 3213 /8 0.45 + 0.825αβ + 0.825 (1-α 2 )(1 -β 2 ) ≤ 3 4 de d 2 /8 ,(2.7)where α, β are defined by |a| = αd and |E e Z | = 1.65 βe d 2 /8 , and we have used that xy + (1 -x 2 )(1 -y 2 ) ≤ 1 for 0 ≤ x, y ≤ 1.

Figure 1 :

 1 Figure 1: Boundaries of S

( 3 )

 3 d (red) for d = 2, 3, 4, 5

  Since diam(3Z/d) ≤ 3, we have by Theorem 2.1 that |E e 3Z/d -1| ≤ e 3 2 /8 -1, and since we are assuming that d ≥ 3, (2.7) gives that |E Ze Z | ≤ 1 4 d 2 s e d 2 s 2 /8 for s ≥ 3/d. Therefore, |E e Z -1| ≤ e 3 2 /8 -1 + se d 2 s 2 /8 = e d 2 /8 -1.

		1
		E Ze sZ ds.
		3/d
	1 3/d	1 4 d 2

Research supported by the Australian Research Council.

(3) d are not always equal. However it could be that the parts of them in the right half-plane are equal.

By considering mixture with an identically-zero random variable, we find that the region S d is star-like from the point 1, and therefore simply connected. Lemma 2.2 applied to functions of the form F (z) = e z-iθ shows that S d and S

(3) d have the same convex hull, and therefore are the same if S

(3)

d is not always convex; there is a shallow indentation on the left side when d = 3, for example. There is also an indentation where the red and black curves in the figure meet.