KAZHDAN CONSTANTS, CONTINUOUS PROBABILITY
MEASURES WITH LARGE FOURIER COEFFICIENTS AND
RIGIDITY SEQUENCES
by

arXiv:1804.01369v1 [math.DS] 4 Apr 2018

Catalin Badea & Sophie Grivaux

To the memory of Jean-Pierre Kahane (1926-2017)

Abstract. — Exploiting a construction of rigidity sequences for weakly mixing dynamical
systems by Fayad and Thouvenot, we show that for every integers p1 , . . . , pr there exists a
continuous probability measure µ on the unit circle T such that
inf

k1 ≥0,...,kr ≥0

|b
µ(pk1 1 . . . pkr r )| > 0.
′

This results applies in particular to the Furstenberg set F = {2k 3k ; k ≥ 0, k′ ≥ 0}, and
disproves a 1988 conjecture of Lyons inspired by Furstenberg’s famous ×2-×3 conjecture.
We also estimate the modified Kazhdan constant of F and obtain general results on rigidity
sequences which allow us to retrieve essentially all known examples of such sequences.

1. Introduction
Denote by T the unit circle T = {λ ∈ C ; |λ| = 1}, by M(T) the set of (finite) complex
Borel measures on T and by P(T) the set of Borel probability measures on T. The Fourier
coefficients of µ ∈ M(T) are defined here as
Z
λn dµ(λ).
µ̂(n) =
T

A measure µ ∈ P(T) is said to be continuous, or atomless, if µ({λ}) = 0 for every λ ∈ T.
We denote the set of continuous probability measures on T by Pc (T). According to a
theorem of Wiener and the Koopman-von Neumann lemma, µ is continuous if and only
if µ̂(n) tends to zero as n tends to infinity along a sequence in N of density one. For
2000 Mathematics Subject Classification. — 43A25, 37A05, 37A25.
Key words and phrases. — Fourier coefficients of continuous measures; non-lacunary semigroups of
integers; Furstenberg Conjecture; rigidity sequences for weakly mixing dynamical systems; Kazhdan subsets
of Z.
This work was supported in part by the project FRONT of the French National Research Agency (grant
ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).
We are grateful to Étienne Matheron for pointing out a simplification of our original proof of Theorem 2.4,
and to Étienne Matheron, Martine Queffélec, Jean-Paul Thouvenot and Benjy Weiss for several interesting
discussions.

2

CATALIN BADEA & SOPHIE GRIVAUX

every µ ∈ P(T), we define µ
e by setting µ
e(A) = µ(Ac ) for every Borel set A ⊆ T, with
e has the property that ν̂(n) = |µ̂(n)|2 ≥ 0 for every
Ac = {λ ; λ ∈ A}. Then ν := µ ∗ µ
n ∈ Z, and ν belongs to Pc (T) as soon as µ does.
A conjecture of Russell Lyons. — Our aim in this paper is to study some nonlacunary sets of positive integers from a Fourier analysis point of view, and to construct
some probability measures which have large Fourier coefficients on such sets. In particular,
we disprove a 1988 conjecture of Lyons [31], called there Conjecture (C4), which reads as
follows:
Lyons’ Conjecture (C4): If S is a non-lacunary semigroup of integers, and if
µ ∈ Pc (T), there exists an infinite sequence (nk )k≥1 of elements of S such that
/ +∞.
/ 0 as k
µ
b(nk )

This conjecture of Lyons is inspired by Furstenberg’s famous conjecture concerning
common invariant probability measures for two commuting automorphisms Tp : λ ✤ / λp
and Tq : λ ✤ / λq of the unit circle T when p and q are two multiplicatively independent
integers (i.e. p and q are not both powers of the same integer). In this setting, Furstenberg’s
conjecture states that the only continuous probability measure on T invariant by both Tp
and Tq is the Lebesgue measure on T. Furstenberg himself proved [22] that if S is any
non-lacunary semigroup of integers (i.e. if S is not contained in any semigroup of the form
{an ; n ≥ 0}, a ≥ 2), the only infinite closed S-invariant subset of T is T itself. See [10]
for an elementary proof of this result and the references mentioned in [15, Chapter 2] for
′
several extensions. Since S = {pk q k ; k, k ′ ≥ 0} is a non-lacunary semigroup whenever
p and q are multiplicatively independent, the only infinite closed subset of T which is
simultaneously Tp -invariant and Tq -invariant is T. Starting with the work of Lyons in
[31], Furstenberg’s conjecture has given rise to an impressive amount of related questions
and results, concerning in particular the dynamics of commuting group automorphisms.
We refer the reader to the papers [35], [14] or [25] for example, as well as to the texts [29],
[26] or [36] for surveys of results related to this conjecture, as well as for perspectives.
As written in [31], conjecture (C4) is a natural version of Furstenberg’s conjecture about
measures, but not involving invariance. If (C4) were true, it would obviously imply an
affirmative answer to the Furstenberg conjecture.
Kazhdan sets and modified Kazhdan constants. — It turns out that Lyons’ conjecture is related to an important property of subsets of Z, namely that of being or not a
Kazhdan subset of Z. Kazhdan subsets Q of a second-countable topological group G are
those for which there exists ε > 0 such that any strongly continuous representation π of
G on a complex separable Hilbert space H admitting a vector x ∈ H with ||x|| = 1 which
is ε-invariant on Q (i.e. supg∈Q ||π(g)x − x|| < ε) has a G-invariant vector. Such an ε is
called a Kazhdan constant for Q, and the supremum of all ε’s with this property is the
Kazhdan constant of Q. Groups with Property (T), also called Kazhdan groups, are those
which admit a compact Kazhdan set. See the book [7] for more on Property (T) and its
numerous important applications.
As suggested in [7, Sec. 7.12], it is of interest to study Kazhdan sets in groups which
do not have Property (T), such as locally compact abelian groups, Heisenberg groups,
etc. See [4] and also [17] for a study of such problems. In the case of the group Z, the
definition above is equivalent to the following one:

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

3

Definition 1.1. — (Kazhdan sets and constants) A subset Q ⊂ Z is said to be a Kazhdan
set if there exists ε > 0 such that any unitary operator U acting on a complex separable
Hilbert space H satisfies the following property: if there exists a vector x ∈ H with
||x|| = 1 such that supn∈Q ||U n x − x|| < ε, then there exists a non-zero vector y ∈ H such
that U y = y (i.e. 1 is an eigenvalue of U ). We will say in this case that (Q, ε) is a Kazhdan
pair. We define the Kazhdan constant of Q as
Kaz(Q) = inf inf sup kU q x − xk,
U kxk=1 q∈Q

where the first infimum is taken over all unitary operators U on H without fixed vectors.
It follows from [7, p. 30] that 0 ≤ Kaz(Q) ≤

√

2 for every Q ⊆ Z.

Several characterizations of Kazhdan subsets of Z were obtained in [4] as consequences
of results applying to a much wider class of groups; self-contained proofs of these characterizations of Kazhdan subsets of Z, involving only classical tools from harmonic analysis,
were obtained in the paper [5]. One of the characterizations of generating Kazhdan sets
obtained in [4, Th. 6.1] (see also [5, Th. 4.12]) runs as follows. Recall that Q is said to be
generating in the group Z if the smallest subgroup containing Q is Z itself.
Theorem 1.2 ([4]). — Let Q be a√generating subset of Z. Then Q is a Kazhdan subset of
Z if and only if there exists ε′ ∈ (0, 2] such that (Q, ε′ ) is a modified Kazhdan pair, that is
any unitary operator V acting on a complex separable Hilbert space H satisfies the following
property: if there exists a vector x ∈ H with ||x|| = 1 such that supn∈Q ||V n x − x|| < ε′ ,
then V has at least one eigenvalue.
We define now the modified Kazhdan constant of Q as
g
Kaz(Q)
= inf inf sup kV q x − xk,
V kxk=1 q∈Q

where the first infimum is taken this time over unitary operators V on H without eigenvalues (that is, with continuous spectra). Therefore
√
g
0 ≤ Kaz(Q) ≤ Kaz(Q)
≤ 2

g
and for every Q ⊆ Z, Kaz(Q) = 0 if and only if Kaz(Q)
= 0 if and only if Q is a nonKazhdan set. The property of being or not a Kazhdan set can also be expressed in terms
of Fourier coefficients of probability measures; see Section 4 for a discussion.
The characterization of Kazhdan subsets of Z obtained by the authors in [4] (see also
[5]) implies that the generating subsets Q of Z which satisfy the property stated in (C4)
(namely that there exists for every µ ∈ Pc (T) an infinite sequence (nk )k≥1 of elements
/ +∞) are exactly the Kazhdan subsets of Z with
/ 0 as k
of Q such that µ
b(nk )
√
√
g
modified Kazhdan constant Kaz(Q)
= 2. Since 2 is the modified Kazhdan constant
√
of Z seen as a subset of itself, 2 is the maximal modified Kazhdan constant, and thus
(C4) can be reformulated as: every generating non-lacunary semigroup
S of integers is
√
a Kazhdan subset of Z with maximal modified Kazhdan constant 2. The relationship
between Furstenberg ×2-×3 conjecture and modified Kazhdan constants can be also seen
directly from Proposition 4.4 below.

CATALIN BADEA & SOPHIE GRIVAUX

4

2. Main results
The first main result of this paper is the following:
Theorem 2.1. — Let p1 , . . . , pr be positive distinct integers and set
E = {pk11 . . . pkr r ; k1 ≥ 0, . . . , kr ≥ 0}.

There exists a continuous probability measure µ on T such that
inf

k1 ≥0,...,kr ≥0

|b
µ(pk11 . . . pkr r )| > 0.

Equivalently,

√

g
Kaz(E)
<

2.

It should be noted that, as conjecture (C4) does not involve invariant measures, we do
not assume in Theorem 2.1 that the integers pj are multiplicatively independent. Notice
also that the statement of Theorem 2.1 is well-known in the lacunary case: if r = 1 it
suffices to consider the classical Riesz product associated to the sequence (pk )k≥0 . In
the non-lacunary case, Theorem 2.1 disproves Conjecture (C4), as well as the related
conjectures (C5) and (C6) of [31] (which are both stronger than (C4)). It applies in
′
particular to the Furstenberg set F = {2k 3k ; k, k′ ≥ 0} and shows the existence of a
measure µ ∈ Pc (T) such that
′
inf
µ
b(2k 3k ) > 0.
′
k,k ≥0

In view of this result, one may naturally wonder for which values of δ ∈ (0, 1) there exists
a measure µ ∈ Pc (T) such that
′

inf
µ
b(2k 3k ) ≥ δ,
′

k,k ≥0

or, equivalently, whether the Furstenberg set F is a Kazhdan set in Z, and if yes, with
which (modified) Kazhdan constant. In this direction, we prove the following result:
′
g ) ≤ 1. More precisely, there
Theorem 2.2. — Let F = {2k 3k ; k, k′ ≥ 0}. Then Kaz(F
exists for every δ ∈ (0, 1/2) a continuous probability measure µ on T with nonnegative
Fourier coefficients such that
′
inf
µ
b(2k 3k ) > δ.
′

k,k ≥0

Rigidity sequences. — Our strategy for proving Theorem 2.1 is to construct measures µ ∈ Pc (T) whose Fourier coefficients tend to 1 along a substantial part of the
set {pk11 . . . pkr r ; k1 ≥ 0, . . . , kr ≥ 0}. In other words, we show that certain large subsets of this set form, when taken in a strictly increasing order, rigidity sequences in the
sense of [8] or [18]. Recall that a dynamical system (X, B, m; T ) on a Borel probability
space is called rigid if there exists a strictly increasing sequence of integers (nk )k≥1 such
/ +∞ for every f ∈ L2 (X, B, m), where UT denotes as
/ 0 as k
that ||UTnk f − f ||
✤
/ f ◦ T associated to T on L2 (X, B, m). Equivalently,
usual the Koopman operator f
−n
/
/
m(T k A △ A)
0 as k
+∞ for every A ∈ B. We say in this case that the system is rigid with respect to the sequence (nk )k≥1 , or that (nk )k≥1 is a rigidity sequence
for (X, B, m; T ). The case where the system (X, B, m; T ) is weakly mixing is particularly
interesting, and is the object of the works [8] and [18]. A strictly increasing sequence
(nk )k≥1 of integers is called a rigidity sequence if there exists a weakly mixing system
which is rigid with respect to (nk )k≥1 .

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

5

Using Gaussian dynamical systems, one can show that (nk )k≥1 is a rigidity sequence
/ +∞.
/ 1 as k
if and only if there exists a measure µ ∈ Pc (T) such that µ
b(nk )
The study of rigidity sequences was initiated in [8] and [18]. Further works on this topic
include the papers [1], [3], [2], [23], [21] and [20] among others. The paper [21] by
Fayad and Thouvenot is especially relevant here: the authors re-obtain a result of Adams
[3], showing that whenever (nk )k≥1 is a rigidity sequence for an ergodic rotation on the
circle, it is a rigidity sequence for a weakly mixing system. The proof of this result in
[3] relies on an involved construction of a suitable weakly mixing system by cutting and
stacking, while the authors of [21] proceed by a direct construction of suitable continuous
/ 1 for some λ = e2iπθ ∈ T with θ ∈ R \ Q,
probability measures: they show that if λnk
/ 1. We obtain the following theorem, which
there exists µ ∈ Pc (T) such that µ
b(nk )
generalizes the result of Fayad and Thouvenot:
Theorem 2.3. — Let (nk )k≥0 be a strictly increasing sequence of integers. Suppose that
the set
/ 1 as k
/ +∞ }
C = {λ ∈ T ; λnk
is dense in T. Then (nk )k≥1 is a rigidity sequence, and there exists a continuous probability
/ 1 as k
/ +∞.
measure µ on T such that µ
b(nk )

This result allows us to retrieve essentially all known examples of rigidity sequences (a
notable exception being the examples of [20]). Notice that C, like every subgroup of the
circle group, is dense in T as soon as it is infinite.
We deduce from Theorem 2.3 the following two-dimensional statement, which is asymmetric and involves a uniformity assumption.
Theorem 2.4. — Let (mk )k≥0 and (nk′ )k′ ≥0 be two strictly increasing sequences of inte/ +∞ as k
/ +∞, and set
/ N be such that ψ(k)
gers. Let also ψ : N
Dψ = {(k, k′ ) ∈ N2 ; 0 ≤ k′ ≤ ψ(k)}.
Suppose that the set
Cψ′ = {λ ∈ T ; λmk nk′

/ 1 as k

/ +∞ , (k, k ′ ) ∈ Dψ }

is dense in T.
/1
Then there exists a continuous probability measure µ on T such that µ
b(mk nk′ )
′
/
+∞ with (k, k ) ∈ Dψ . In other words, there exists for every ε > 0 an integer
as k
k0 such that |b
µ(mk nk′ ) − 1| < ε for every (k, k′ ) ∈ N2 with k ≥ k0 and 0 ≤ k′ ≤ ψ(k).
Remark that the assumption of Theorem 2.4 is in particular satisfied if the set
C ′ = {λ ∈ T ; λmk nk′

/ 1 as k

/ +∞ uniformly in k ′ }

is dense in T.
Theorem 2.1 is obtained by first observing that the set {pk11 . . . pkr r ; p1 ≥ 0, . . . , pr ≥ 0}
can be split into r sets to which Theorem 2.4 applies, and then considering a convex
combination of the continuous measures obtained in this way.
In order to prove Theorem 2.2, one needs to refine the statement of Theorem 2.4,
and to show that the sequences (mk nk′ )k,k′ ≥0 satisfying the assumption of Theorem 2.4
actually give rise to non-Kazhdan subsets of Z. More precisely, we prove the following
strengthenings of Theorems 2.3 and 2.4 respectively:

CATALIN BADEA & SOPHIE GRIVAUX

6

Theorem 2.5. — Under the assumption of Theorem 2.3, there exists for every ε > 0 a
/ +∞ and sup
/ 1 as k
µ(nk ) − 1| < ε. In
measure µ ∈ Pc (T) such that µ
b(nk )
k≥0 |b
particular, {nk ; k ≥ 0} is a non-Kazhdan subset of Z.
Theorem 2.6. — Under the assumption of Theorem 2.4, there exists for every ε > 0 a
/ 1 as k
/ +∞ with (k, k ′ ) ∈ Dψ , and
measure µ ∈ Pc (T) such that µ
b(mk nk′ )
sup

k≥0, 0≤k ′ ≤ψ(k)

|b
µ(mk nk′ ) − 1| < ε.

In particular, {mk nk′ ; k ≥ 0, 0 ≤ k′ ≤ ψ(k)} is a non-Kazhdan subset of Z.
Organization of the paper. — The paper is structured as follows. We give in Section 3
the proof of Theorems 2.3 and 2.4, which are very much inspired from the paper [21]. In
Section 4, we recall a characterization of generating Kazhdan subsets of Z from [4], and
detail the links between several natural constants involved in this characterization. We
explain in particular why the generating subsets of Z which satisfy the property
√ stated in
(C4) are exactly the Kazhdan subsets of Z with modified Kazhdan constant 2. We then
prove Theorems 2.5 and 2.6. Section 5 is devoted to applications: we prove Theorems 2.1
and 2.2, and show how to retrieve many examples of rigidity sequences, using Theorems
2.3 and 2.4. We also provide an application of Theorem 2.2 to the study of the size
of the exceptional set of values θ ∈ R for which the sequence (nk θ)k≥0 is not almost
uniformly distributed modulo 1 with respect to a (finite) complex Borel measure ν ∈ M(T),
where (nk )k≥0 denotes the Furstenberg sequence: we show that this exceptional set is
uncountable, thus providing a new example of a sublacunary sequence with uncountable
exceptional set for (almost) uniform distribution.
3. Proof of Theorems 2.3 and 2.4
Given two integers a < b, we will when the context is clear denote by [a, b] the set of
integers k such that a ≤ k ≤ b.
Proof of Theorem 2.3. — The general strategy of the proof is the following: we construct a
sequence (λi )i≥1 of pairwise distinct elements of C, as well as a strictly increasing sequence
of integers (Np )p≥0 , such that the measures
p

−p

µp = 2

2
X
i=1

δ{λi } ,

p≥0

satisfy certain properties stated below. At step p, we determine the elements λi for
2p−1 < i ≤ 2p as well as the integer Np in such a way that λ1 = 1, N0 = 0, and
(1) for every p ≥ 1, every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ],
Z
|λnk − 1| dµp (λ) < 2−(j−1) ;
T

(2) for every p ≥ 1, every q ∈ [0, p − 1], l ∈ [1, 2p−q ), r ∈ [1, 2q ],
|λl2q +r − λr | < ηq

where ηq =

1
inf 1≤i<j≤2q |λi − λj | for every q ≥ 1, and η0 = 1;
4

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

7

(3) for every p ≥ 1 and every k ≥ Np ,
Z
|λnk − 1| dµp (λ) < 2−(p+1) .
T

Remark that property (2) implies that the sequence (λi )i≥1 satisfies
(4) for every q ≥ 0, every l ≥ 0, and every r ∈ [1, 2q ],
|λl2q +r − λr | < ηq .

Suppose that the sequences (λi )i≥1 and (Np )p≥0 have been constructed so as to satisfy
(1), (2), and (3) above, and let µ be a w∗ -limit point of the sequence (µp )p≥0 in P(T).
Claim 3.1. — We have µ
b(nk )

/ 1 as k

/ +∞.

Proof. — For every k ≥ 0, denote by jk ≥ 0 the unique integer j such that k ∈ [Nj , Nj+1 ).
For every p > jk , we have by (1)
Z
Z
nk
−(jk −1)
|λnk − 1| dµ(λ) ≤ 2−(jk −1) .
|λ − 1| dµp (λ) < 2
so that
Z
T
T
/ 0, i.e. µ
/ +∞,
/1.
/ +∞ as k
|λnk − 1| dµ(λ)
b(nk )
Since jk
T

Claim 3.2. — The probability measure µ is continuous.

Proof. — Fix q ≥ 1, and consider for every r ∈ [1, 2q ] the two arcs Γr and ∆r of T defined
by
3
Γr = {λ ∈ T ; |λ − λr | ≤ ηq } and ∆r = {λ ∈ T ; |λ − λr | < ηq }.
2
The 2q arcs ∆r are pairwise disjoint. Indeed, for every r, r ′ ∈ [1, 2q ] with r 6= r ′ , every
λ ∈ ∆r and every λ′ ∈ ∆r′ , we have by the definition of ηq that
|λ − λ′ | ≥ |λr − λr′ | − 3ηq ≥ 4ηq − 3ηq = ηq > 0.

So ∆r and ∆r′ do not intersect.

Let us next estimate, for every r ∈ [1, 2q ] and every p ≥ q, the quantity µp (Γr ). We
have
µp (Γr ) = 2−p × #{i ∈ [1, 2p ] ; λi ∈ Γr }.
Every i ∈ [1, 2p ] can be written as i = l2q + s for some l ≥ 0 and s ∈ [1, 2q ]. By (4), λi
belongs to Γs . Since the arcs ∆r′ , r ′ ∈ [1, 2q ], are pairwise disjoint, it follows that
Also,

µp (∆r ) = µp (Γr ) = 2−p × #{i ∈ [1, 2p ] ; i ≡ r mod 2q } ≤ 2−q .
µp

2q
[

r=1


Γr = 1.

Since the arcs Γr are closed while the arcs ∆r are open, going to the limit as p goes to
infinity yields that µ(∆r ) ≤ 2−q for every r ∈ [1, 2q ] and
µ

2q
[

r=1


Γr = 1.

If λ ∈ T is such that µ({λ}) > 0, there exists an r ∈ [1, 2q ] such that λ ∈ Γr ⊂ ∆r . So
µ({λ}) ≤ µ(∆r ) ≤ 2−q , a contradiction if q is sufficiently large. It follows that the measure
µ is continuous.

CATALIN BADEA & SOPHIE GRIVAUX

8

By Claims 3.1 and 3.2, it suffices to construct (λi )i≥0 and (Np )p≥0 satisfying (1), (2),
and (3) in order to prove Theorem 2.3. For p = 0, we set λ1 = 1 and N0 = 0, so that
µ0 = δ{1} .
For p = 1, we choose λ2 ∈ C distinct from λ1 with |λ2 − λ1 | < 1. Since µ1 = 12 (δ{1} +
δ{λ2 } ), we have
Z
1
|λnk − 1| dµ1 (λ) = |λn2 k − 1| ≤ 1 < 2 for every k ≥ 0.
2
T
Hence property (1) is satisfied whatever the choice of N1 . Since η0 = 1 and |λ2 − λ1 | < 1,
property (2) is satisfied. It remains to choose N1 in such a way that property (3) is
satisfied. Since λ2 belongs to C,
Z
1
/ 0 as k
/ +∞,
|λnk − 1| dµ1 (λ) = |λn2 k − 1|
2
T
so we can choose N1 so large that
Z
|λnk − 1| dµ1 (λ) < 2−2 for every k ≥ N1 .
T

This terminates the construction for p = 1.
Suppose now that the construction has been carried out until step p, i.e. that the
quantities λi , i ∈ [1, 2p ], and Nl , l ∈ [0, p], have been constructed satisfying (1), (2), and
(3).
We construct by induction on s ∈ [1, 2p ] elements λ2p +s of C, measures µp,s ∈ P(T),
and integers Np,s in such a way that the elements λi , i ∈ [1, 2p+1 ], are all distinct, Np <
Np,1 < · · · < Np,2p , and the following three properties are satisfied:
(a) for every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ],
Z
|λnk − 1| dµp,s (λ) < 2−(j−1) ;
T

(b) for every k ≥ Np ,
(c) for every k ≥ Np,s,

Z
Z

T

|λnk − 1| dµp,s (λ) < 2−(p−1) ;

T

|λnk − 1| dµp,s (λ) < 2−(p+2) .

Let us start with the construction of λ2p +1 . By density of C, one can choose λ2p +1
distinct from all the elements λi , i ∈ [1, 2p ], with |λ2p +1 − λ1 | arbitrarily small. Consider
the measure

µp,1 = µp + 2−(p+1) δ{λ2p +1 } − δ{λ1 }
−(p+1)

=2



p

−p

δ{λ1 } + δ{λ2p +1 } + 2

2
X

δ{λi }

i=2

obtained by splitting the point mass δ{λ1 } appearing in the expression of µp into 12 (δ{λ1 } +
δ{λ2p +1 } ). We have for every k ≥ 0
Z
Z
nk
|λnk − 1| dµp (λ) + 2−(p+1) |λn2pk+1 − λn1 k |.
|λ − 1| dµp,1 (λ) ≤
(5)
T

T

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

9

If |λ2p +1 − λ1 | is sufficiently small, we have by (1) that
Z
|λnk − 1| dµp,1 (λ) < 2−(j−1)
T

for every j ∈ [0, p−1] and every k ∈ [Nj , Nj+1 ], i.e. that (1) still holds true for the measure
µp,1 . Also (5) and (3) imply that for every k ≥ Np ,
Z
|λnk − 1| dµp,1 (λ) < 2−(p+1) + 2−p < 2−(p−1) ,
T

which is (b). Since all the elements λi , i ∈ [1, 2p + 1], belong to C, there exists Np,1 > Np
such that
Z
|λnk − 1| dµp,1 (λ) < 2−(p+2) for every k ≥ Np,1 .
T

Properties (a), (b), and (c) are thus satisfied for s = 1. Suppose now that λ2p +s′ , µ2p +s′ ,
and N2p +s′ have been constructed for s′ < s. Let λ2p +s ∈ C \ {λ1 , . . . , λ2p +s−1 } be very
close to λs , and set

µp,s = µp,s−1 + 2−(p+1) δ{λ2p +s } − δ{λs }
(this time, the mass point δ{λs } appearing in µp is split as 12 (δ{λs } + δ{λ2p +s } ). Since, for
every k ≥ 0,
(6)

Z

T

|λ

nk

− 1| dµp,s (λ) ≤

Z

T

|λnk − 1| dµp,s−1 (λ) + 2−(p+1) |λn2pk+s − λns k |,

the induction assumption implies that (a) holds true provided |λ2p +s − λs | is sufficiently
small. As to (b), we have to consider separately the cases Np ≤ k < Np,s−1 and k ≥ Np,s−1 .
If |λ2p +s − λs | is sufficiently small,
Z
|λnk − 1| dµp,s (λ) < 2−(p−1) for every Np ≤ k < Np,s−1 .
T

By property (c) at step s − 1 and (6),
Z
|λnk − 1| dµp,s (λ) < 2−(p+2) + 2−p < 2−(p−1)
T

for every k ≥ Np,s−1 . Hence (b) is satisfied at step s. Property (c) is satisfied if Np,s is
chosen sufficiently large since all the elements λi , i ∈ [1, 2p + s], belong to C.

Let us now set µp+1 = µp,2p and Np+1 = Np,2p . It remains to check that with these
choices of λi , i ∈ [1, 2p+1 ], µp+1 and Np+1 , properties (1), (2), and (3) are satisfied.
By (a), property (1) is satisfied for every j ∈ [0, p − 1]. The case where j = p follows
from (b). So (1) is true. Property (3) follows immediately from (c), so it only remains to
check (2).

Fix q ∈ [0, p], l ∈ [1, 2p+1−q ) and r ∈ [1, 2q ]. Consider first the case where q = p. In
this case l = 1, and the quantities under consideration have the form |λ2p +r − λr |, with
r ∈ [1, 2p ]. One can ensure in the construction that |λ2p +r − λr | < ηp for every r ∈ [1, 2p ]
and then (2) holds true for q = p.
Suppose then that q ∈ [0, p − 1], and write l as l = l′ + ε2p−q with ε ∈ {0, 1} and l′ ∈
[1, 2p−q ). Then l2q +r = l′ 2q +r+ε2p . Set s = l′ 2q +r. Then 1 ≤ s ≤ (2p−q −1)2q +2q = 2p ,
i.e. s ∈ [1, 2p ]. We have
|λl2q +r − λr | ≤ |λs+ε2p − λs | + |λl′ 2q +r − λr |.

CATALIN BADEA & SOPHIE GRIVAUX

10

If ε = 0, the first term is zero; if ε = 1, it is equal to |λ2p +s − λs |, which can be assumed
to be as small as we wish in the construction. As to the second term, it is less than ηq by
property (2) at step p, since l′ ∈ [1, 2p−q ) and r ∈ [1, 2q ] with q ∈ [0, p − 1]. We can thus
ensure that
|λl2q +r − λr | < ηq

for every q ∈ [0, p], l ∈ [1, 2p+1−q ), and r ∈ [1, 2q ]. This proves that property (2) is satisfied
at step p + 1, and this concludes the proof of Theorem 2.3.
Theorem 2.4 is now a formal consequence of Theorem 2.3.
Proof of Theorem 2.4. — Recall that Dψ = {(k, k′ ) ∈ N2 ; 0 ≤ k′ ≤ ψ(k)} and
Cψ′ = {λ ∈ T ; λmk nk′

/ 1 as k

/ +∞ , (k, k ′ ) ∈ Dψ }.

Order the set {mk nk′ ; (k, k′ ) ∈ Dψ } as a strictly increasing sequence (pl )l≥1 of integers.
Since there exists for every integer k1 ≥ 0 an integer l1 ≥ 0 such that
{pl ; l ≥ l1 } ⊆ {mk nk′ ; (k, k′ ) ∈ Dψ , k ≥ k1 },

/ +∞. By Theorem 2.3
/ 1 as l
every element λ ∈ Cψ′ has the property that λpl
/ 1 as l
/ +∞.
applied to the sequence (pl )l≥1 , there exists µ ∈ Pc (T) such that µ
b(pl )
Using this time the fact that there exists for every integer l2 ≥ 0 an integer k2 ≥ 0 such
that
{mk nk′ ; (k, k′ ) ∈ Dψ , k ≥ k2 } ⊆ {pl ; l ≥ l2 },

we deduce that µ
b(mk nk′ )

/ 1 as k

/ +∞ with (k, k ′ ) ∈ Dψ .

Remark 3.3. — Suppose that the set
C ′ = {λ ∈ T ; λmk nk′

/ 1 as k

/ +∞ uniformly in k ′ }

is dense in T. It is natural to wonder whether there exists a measure µ ∈ Pc (T) such that
/ 1 as k
/ 1 uniformly in k ′ . The following example shows that it is not
µ
b(mk nk′ )
k
the case: set mk = 2 and nk′ = k′ for every k, k′ ≥ 0. The set
C ′ = {λ ∈ T ; λmk nk′

/ 1 as k

/ +∞ uniformly in k ′ }

contains all 2k -th roots of 1, and so is dense in T. Suppose that µ ∈ P(T) is such that
/ 1 as k
/ +∞ uniformly in k ′ . Then there exists an integer k0 ≥ 1 such that
µ
b(2k k′ )
k
′
|b
µ(2 0 k )| ≥ 1/2 for every k′ ≥ 0. Consider the measure ν = T2k0 (µ). Since νb(n) = µ
b(2k0 n)
k
for every n ∈ Z, ν cannot be continuous. Also, ν({λ0 }) = µ({λ ∈ T ; λ2 0 = λ0 }) for every
λ0 ∈ T, and so the measure µ itself cannot be continuous.
So the conclusion of Theorem 2.4 seems to be essentially optimal.
4. Some non-Kazhdan subsets of Z
4.1. Kazhdan constants and Fourier coefficients of probability measures. —
We begin this section by recalling a characterization of generating Kazhdan subsets of
Z, obtained in [4, Th. 6.1] (see also [5, Th. 4.12]) and presenting some facts concerning
the (modified) Kazhdan constants of such sets. We state it here in a slightly modified
way (condition (ii) is not exactly the same as in [5, Th. 4.12]), and include a discussion
concerning the links between the various constants appearing in the equivalent conditions.

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

11

Theorem 4.1. — Let Q be a generating subset of Z. Then Q is a Kazhdan subset of Z
if and only if one of the following equivalent assertions holds true:
√
(i) there exists ε ∈ (0, 2) such that (Q, ε) is a modified Kazhdan pair. Equivalently,
g
Kaz(Q)
≥ ε;
(ii) there exists γ ∈ (0, 1) such that any measure µ ∈ P(T) with supn∈Q (1 − ℜe µ
b(n)) < γ
has a discrete part;
(iii) there exists δ ∈ (0, 1) such that any measure µ ∈ P(T) with inf n∈Q |b
µ(n)| > δ has a
discrete part.
Moreover:
√
– (i) is satisfied for ε ∈ (0, 2) if and only if (ii) is satisfied for γ = ε2 /2;
√
– if (ii) is satisfied for γ ∈ (0, 1), (iii) is satisfied for δ = 1 − γ, while if (iii) is
satisfied for δ ∈ (0, 1), (ii) is satisfied for γ = 1 − δ;
q
√
– hence if (i) is satisfied for ε ∈ (0, 2), (iii) is satisfied for δ = 1 − ε2 /2 , while if
p
(iii) is satisfied for δ ∈ (0, 1), (i) holds true for ε = 2(1 − δ).

We prove briefly here the statement concerning the relations between the constants ε,
γ, and δ appearing in (i), (ii), and (iii) respectively, following [4] and [5].
√
Proof. — Suppose that (i) is satisfied for ε ∈ (0, 2), and let µ ∈ P(T). Consider the
unitary operator U = Mλ of multiplication by λ on L2 (T, µ). Let f be the function
constantly equal to 1. Then ||U n f − f ||2 = 2(1 − ℜe µ
b(n)). If supn∈Q (1 − ℜe µ
b(n)) < ε2 /2,
g
U has an eigenvalue since Kaz(Q)
≥ ε, and so µ has a discrete part.
Conversely, suppose that (ii) is satisfied for γ ∈ (0, 1). Let U be a unitary operator on
a separable Hilbert space H, and let x ∈ H with ||x|| = 1 be such that
p
sup ||U n x − x|| < 2γ.
n∈Q

The proof of [5, Th. 4.6] shows then that there exists µ ∈ P(T) such that
2 sup (1 − ℜe µ
b(n)) = sup ||U n x − x||2 < 2γ.
n∈Q

n∈Q

So supn∈Q (1 − ℜe µ
b(n)) < γ. By (ii), µ has a discrete part, and so U has an eigenvalue.
√
g
Hence Kaz(Q) ≥ 2γ.
Suppose next that
√ property (ii) is satisfied for γ ∈ (0, 1). Let µ ∈ P(T) be such that
inf n∈Q |b
µ(n)| > 1 − γ. Set ν = µ ∗ µ
e. Then inf n∈Q νb(n) > 1 − γ. It follows that
supn∈Q (1 − νb(n)) < γ, and ν has a discrete part. So µ itself has a discrete part.

Lastly, suppose that (iii) is satisfied for δ ∈ (0, 1). Let µ ∈ P(T) be a measure satisfying
supn∈Q (1−ℜe µ
b(n)) < 1 − δ. Then inf n∈Q |b
µ(n)| ≥ inf n∈Q ℜe µ
b(n) > δ, so µ has a discrete
part.
Remark 4.2. — Given a subset Q of Z, one can prove, using the spectral theorem for
unitary operators, that the following assertions are equivalent (see [5, Th. 4.6]):
√
(i’) Q is a Kazhdan subset of Z, i.e. there exists ε ∈ (0, 2) such that (Q, ε) is a Kazhdan
pair;
(ii’) there exists γ ∈ (0, 1) such that any measure µ ∈ P(T) with supn∈Q (1 − ℜe µ
b(n)) < γ
is such that µ({1}) > 0.

CATALIN BADEA & SOPHIE GRIVAUX

12

√
Moreover (i’) holds true for a certain constant ε ∈ (0, 2) (i.e. Kaz(Q) ≥ ε) if and only if
(ii’) holds true for γ = ε2 /2.
It is interesting to note that these two conditions (i’) and (ii’) are not equivalent to the
natural version (iii’) of (iii) (namely, that there exists δ ∈ (0, 1) such that any measure
µ ∈ P(T) with inf n∈Q |b
µ(n)| > δ satisfies µ({1}) > 0). Indeed, (iii’) is satisfied for any
Dirac mass δ{λ} , λ ∈ T. The proof that (ii) implies (iii) in Theorem 4.1 above uses in a
crucial way the fact that if µ ∈ P(T) is such that µ ∗ µ
e has a discrete part, µ itself has a
discrete part. But µ ∗ µ
e may very well satisfy µ ∗ µ
e({1}) > 0 while µ({1}) = 0, and so (ii’)
does not imply (iii’).
Theorem 4.1 is related to Conjecture (C4) in the following way:
Corollary 4.3. — Let Q be a generating subset of Z. The following assertions are equivalent:
√
g
(α) Q is a Kazhdan subset of Z with Kaz(Q)
= 2;
(β) any measure µ ∈ Pc (T) satisfies inf n∈Q |b
µ(n)| = 0;
(γ) any measure µ ∈ Pc (T) satisfies lim inf |n|→+∞ |b
µ(n)| = 0.
n∈Q

Proof. — The equivalence between (α) and (β) follows immediately from Theorem 4.1.
So only the implication (β)=⇒(γ) requires a proof. Suppose that any µ ∈ Pc (T) satisfies inf n∈Q |b
µ(n)| = 0. We want to show that the conclusion can be reinforced into
lim inf |n|→+∞ |b
µ(n)| = 0. Let ρ ∈ Pc (T) be a Rajchman measure with positive coeffin∈Q

cients, that is such that lim|n|→+∞ ρb(n) = 0 and ρb(n) > 0 for every n ∈ Z. Consider
the measure ν = (µ ∗ µ
e + ρ)/2. It is continuous and satisfies ν(n) > 0 for every n ∈ Z.
Since inf n∈Q νb(n) = 0 and ν(n) > 0 for every n ∈ Z, lim inf |n|→+∞ νb(n) = 0. Hence
µ(n)|2 = 0, and the conclusion follows.
lim inf |n|→+∞ |b

n∈Q

n∈Q

So Conjecture (C4) is equivalent to the statement
that any non-lacunary semigroup
√
of integers has modified Kazhdan constant 2. We can also estimate the Fourier coefficients of a continuous probability measure on T which is T2 - and T3 -invariant in terms
of the modified Kazhdan constant of the Furstenberg set. Notice that Proposition 4.4 is
meaningful only if κ̃ > 0.
′

g ). Let µ be a
Proposition 4.4. — Let F = {2k 3k ; k, k ′ ≥ 0} and set κ̃ = Kaz(F
continuous probability measure on T which is T2 - and T3 -invariant. Then

κ̃2
for every j ∈ Z \ {0}.
2
Proof of Proposition 4.4. — Set, for every j ∈ Z \ {0}, µj = Tj µ. Then µj is a continuous
′
measure which satisfies µ̂j (2k 3k ) = µ̂(j) for every k, k′ ≥ 0 It follows that if δ ∈ (0, 1)
is such
p that (iii) of Theorem 4.1 is satisfied, δ ≥ |µ̂(j)|. Hence, by Theorem 4.1 again,
κ̃ ≤ 2(1 − |µ̂(j)|).
|µ̂(j)| ≤ 1 −

Remark 4.5. — Although a generating subset Q of Z is a Kazhdan set if and only if
g
Kaz(Q)
> 0, there is no link between the Kazhdan constant and the modified Kazhdan
constant of Q. Indeed,
there exist Kazhdan subsets Q of Z with maximal modified con√
g
stant Kaz(Q) = 2 and arbitrarily small Kazhdan constant Kaz(Q). This relies on the
following observation, which can be extracted from the proof of [5, Th 7.1] and results
from Proposition 5.9 below.

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

13

Proposition 4.6. — Let (nk )k≥0 be a strictly increasing sequence of integers with n0 = 1
such that (nk θ)k≥0 is uniformly distributed modulo 1 for every θ ∈ R \ D, where D is
countable subset of R. Then the set Q = {nk ; k ≥ 0} is a Kazhdan subset of Z which
√
g
satisfies Kaz(Q)
= 2.

Consider, for every integer p ≥ 2, the set Qp = p N + 1. By Proposition 4.6, Qp is a
√
g p ) = 2. But the measure µ = δ 2iπ/p satisfies
Kazhdan subset of Z with Kaz(Q
{e
}
Hence Kaz(Qp ) ≤

p

sup (1 − ℜe µ
b(n)) = cos(2π/p).

n∈Qp

2 cos(2π/p), which can be arbitrarily small if p is sufficiently large.

4.2. Proofs of Theorems 2.5 and 2.6. — We only sketch the proof of Theorem 2.5
(Theorem 2.6 is a formal consequence of it).
Proof. — Fix ε ∈ (0, 1/2). Using the notation of the proof of Theorem 2.3, it suffices to
construct the measures µp , p ≥ 0, in such a way that they satisfy the assertions
(1ε ) for every p ≥ 1, every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ],
Z
|λnk − 1| dµp (λ) < 3ε(1 − ε)j ;
T

(2) for every p ≥ 1, every q ∈ [0, p − 1], l ∈ [1, 2p−q ), r ∈ [1, 2q ],
|λl2q +r − λr | < ηq

1
inf 1≤i<j≤2q |λi − λj | for every q ≥ 1, and η0 = 1;
4
(3ε ) for every p ≥ 1 and every k ≥ Np ,
Z
|λnk − 1| dµp (λ) < ε(1 − ε)p+1 ;
where ηq =

T

(4ε ) µp ({λi }) ≤ (1 − ε)p for every i ∈ [1, 2p ].

Then any w∗ -limit point µ of (µp )p≥0 will be a continuous measure which simultaneously
/ 1 as k
/ +∞ and sup
satisfies µ
b(nk )
µ(nk )−1| ≤ 3ε. The main difference with
k≥0 |b
the proof of Theorem 2.3 is that the measures µp will be defined as
p

p

µp =

2
X

(p)
ai δ{λi }

for some suitable weights

api

> 0 with

2
X

(p)

ai

= 1.

i=1

i=1

For p = 0, we set λ1 = 1, N0 = 0, and µ0 = δ{1} .
For p = 1, we choose λ2 ∈ C \ {λ1 } with |λ2 − λ1 | < 1 and set µ1 = (1 − ε)δ{1} + εδ{λ2 } .
We have for every k ≥ 0
Z
|λnk − 1| dµ1 (λ) = ε|λn2 k − 1| ≤ 2ε < 3ε.
T

Since |λ2 − λ1 | < 1, (2) is true. If N1 is chosen sufficiently large, µ1 satisfies properties
(1ε ) and (3ε ). Moreover, µ1 ({1}) = 1 − ε and µ2 ({λ2 }) = ε < 1 − ε, so (4ε ) is true.
Suppose now that the construction has been carried out until step p. We can then
construct by induction on s ∈ [1, 2p ] measures µp,s which satisfy

CATALIN BADEA & SOPHIE GRIVAUX

14

(aε ) for every j ∈ [0, p − 1] and every k ∈ [Nj , Nj+1 ],
Z
|λnk − 1| dµp,s (λ) < 3ε(1 − ε)j ;
T

(bε ) for every k ≥ Np ,
(cε ) for every k ≥ Np,s,

Z
Z

T

T

|λnk − 1| dµp,s (λ) < 3ε(1 − ε)p ;

|λnk − 1| dµp,s (λ) < 3ε(1 − ε)p+2 ;

(dε ) µp,s ({λi }) = µp ({λi }) for every i ∈ (s, 2p ] and µp,s ({λi }) ≤ (1 − ε)p+1 for every
i ∈ [1, s] ∪ [2p + 1, 2p + s].
We define µp,1 as
µp,1 = µp + µp ({1}) ε (δ{λ2p +1 } − δ{λ1 } )
p

= µp ({1}) (1 − ε) δ{λ1 } +

2
X
i=2

µp ({λi }) δ{λi } + µp ({1}) ε δ{λ2p +1 } ,

where λ2p +1 ∈ C \ {λ1 , . . . , λ2p } is such that |λ2p +1 − λ1 | = |λ2p +1 − 1| is very small. Then
for every k ≥ 0,
Z
Z
nk
|λnk − 1| dµp (λ) + (1 − ε)p ε |λn2pk+1 − λn1 k |.
|λ − 1| dµp,1 (λ) ≤
(7)
T

T

It follows that (aε ) holds true for µp,1 , provided that |λ2p +1 −λ1 | is sufficiently small. Also,
we have by (7) and (3ε ) that for every k ≥ Np ,
Z
|λnk − 1| dµp,1 (λ) < ε (1 − ε)p+1 + 2ε (1 − ε)p < 3ε (1 − ε)p
T

so that (bε ) holds true. If Np,1 is sufficiently large, (cε ) is true. Property (dε ) follows from
the expression of µp,1 , since µp,1 ({1}) = µp ({1}) (1 − ε) ≤ (1 − ε)p+1 and µp,1 ({λ2p +1 }) =
µp ({1}) ε ≤ ε (1 − ε)p ≤ (1 − ε)p+1 by (4ε ).

Supposing now that s ≥ 2 and that the construction has been carried out for every
s′ < s, we choose λ2p +s ∈ C \ {λ1 , . . . , λ2p +s−1 } very close to λs , and set
µp,s = µp,s−1 + µp,s−1 ({λs }) ε (δ{λ2p +s } − δ{λs } ).

Then, since µp,s−1 ({λs }) = µp ({λs }) ≤ (1 − ε)p by (dε ) and (4ε ), we have for every k ≥ 0
Z
Z
nk
|λnk − 1| dµp,s−1 (λ) + (1 − ε)p ε |λn2pk+s − λns k |.
|λ − 1| dµp,s (λ) ≤
(8)
T

T

Thus (aε ) holds true provided that |λ2p +s − λs | is sufficiently small. In order to prove (bε ),
suppose first that Np ≤ k < Np,s−1 . Then, if |λ2p +s − λs | is small enough, we have by (8)
and (bε ) for s − 1 that
Z
|λnk − 1| dµp,s (λ) < 3ε (1 − ε)p for every Np ≤ k < Np,s−1 .
T

If k ≥ Np,s−1 we have by (cε ) at step s − 1 and (8) that
Z
|λnk − 1| dµp,s (λ) < ε (1 − ε)p+2 + 2ε (1 − ε)p < 3ε (1 − ε)p .
T

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

15

So (bε ) is satisfied at step s. Property (cε ) is true if Np,s is chosen sufficiently large.
As to property (dε ), we have µp,s({λi }) = µp,s−1 ({λi }) for every i distinct from λs and
λ2p +s . So µp,s ({λi }) = µp ({λi }) for every i ∈ (s, 2p ], and µp,s ({λi }) ≤ (1 − ε)p+1 for every
i ∈ [1, s) ∪ [2p + 1, 2p + s). Also
µp,s ({λs }) = µp,s−1 ({λs }) (1 − ε) = µp ({λs }) (1 − ε) ≤ (1 − ε)p+1 ,

while µp,s({λ2p +s }) = µp,s−1 ({λs }) ε ≤ (1 − ε)p+1 . So (dε ) holds true at step s. This
terminates the construction of the measures µp,s .
We then set µp+1 = µp,2p and Np+1 = Np,2p and check as in the proof of Theorem
2.3 that properties (1ε ), (2), and (3ε ) are satisfied. Since by (dε ) for s = 2p we have
µp,2p ({λi }) ≤ (1 − ε)p+1 for every i ∈ [1, 2p+1 ], property (4ε ) is satisfied as well. This
terminates the construction of the measures µp , and proves Theorem 2.5.
We plan to come back to the study of the links between Kazhdan sets and rigidity
sequences in a forthcoming preprint.
As a direct corollary of Theorems 2.5 and 2.6, we obtain
Corollary 4.7. — For any function ψ : N
the sets
′

{2k 3k ; k ≥ 0, 0 ≤ k′ ≤ ψ(k)}

and

/ N with ψ(k)

/ +∞ as k

/ +∞,

′

{2k 3k ; k′ ≥ 0, 0 ≤ k ≤ ψ(k′ )}

are non-Kazhdan sets in Z.
′

It seems to be unknown whether the Furstenberg set {2k 3k ; k, k′ ≥ 0} is a Kazhdan
set in Z (see Question 5.1 below).
5. Applications
5.1. Proof of Theorem 2.1. — Our first and main application of Theorem 2.4 is
Theorem 2.1, which solves in particular Conjecture (C4) and shows that the invariance
assumption on the measure is indeed essential in the statement of Furstenberg’s ×2 -×3
conjecture.
Proof of Theorem 2.1. — If r = 1, Theorem 2.1 claims the existence, for every integer
p ≥ 2, of a measure µ ∈ Pc (T) such that inf k≥0 |b
µ(pk )| > 0. As mentioned in Section 2,
this statement is well-known: it suffices to consider the classical Riesz product associated
to the sequence (pk )k≥0 . One can also show, either as in [8] or [18], or as an application
of Theorem 2.3, that (pk )k≥0 is a rigidity sequence, so that there exists µ ∈ Pc (T) with
/ +∞.
/ 1 as k
µ
b(pk )
Suppose now that r ≥ 2, and consider, for every fixed index 1 ≤ j ≤ r, the set
−l

Cj′ = {e2iπnpj ; n, l ≥ 0}
of roots of all powers of pj . It is dense in T, and has the following property: there exists
k1 k2
p2 ... pkr r

for every λ ∈ Cj′ an integer lj such that λp1
1 ≤ i ≤ r with i 6= j. Hence
sup
ki ≥0
1≤i≤r, i6=j

k1

λp 1

... pkr r

−1

/0

= 1 for every kj ≥ lj and ki ≥ 0,

as kj

/ +∞.

CATALIN BADEA & SOPHIE GRIVAUX

16

Consider the two sequences (mk )k≥0 and (nk′ )k′ ≥0 obtained by setting mk = pkj , k ≥ 0,
and ordering the set
 k1
kj−1 kj+1
p1 . . . pj−1
pj+1 . . . pkr r ; ki ≥ 0, 1 ≤ i ≤ r with i 6= j

/ N be a strictly increasing
as a strictly increasing sequence (nk′ )k′ ≥0 , and let ψ : N
function such that
 k1
kj−1 kj+1
p1 . . . pj−1
pj+1 . . . pkr r ; 0 ≤ ki ≤ k, 1 ≤ i ≤ r with i 6= r

is contained in the set {nk′ ; 0 ≤ k′ ≤ ψ(k)} for every k ≥ 0. By Theorem 2.4, there exists
/ 1 as kj
/ +∞ with 0 ≤ ki ≤ kj ,
bj (pk11 . . . pkr r )
a measure µj ∈ Pc (T) such that µ
1 ≤ i ≤ r with i 6= j. Replacing, for every 1 ≤ j ≤ r, µj by µj ∗ µ
ej , we can suppose
without loss of generality that µ
bj (n) ≥ 0 for every n ∈ Z.
Let now ρ ∈ Pc (T) be such that ρ(n) > 0 for every n ∈ Z, and set
r

1 X
µj + ρ .
µ=
r+1
j=1

Then µ is a continuous probability measure on T with µ
b(n) > 0 for every n ∈ Z. Moreover,
we have

1
/ +∞.
(9)
lim inf µ
b pk11 pk22 . . . pkr r ≥
as max(k1 , . . . , kr )
r+1
(l)

(l)

Indeed, if (k1 , . . . , kr )l≥1 is an infinite sequence of elements of Nr , one can extract from
(l)
(l)
it a sequence (still denoted by (k1 , . . . , kr )l≥1 ) with the following property: there exists
(l)
(l)
1 ≤ j ≤ r such that ki ≤ kj for every 1 ≤ i ≤ r. Then
(l)
(l)
(l) 
(l) 
1
1
k
k
lim inf µ
b p11 . . . pkr r ≥
bj p11 . . . pkr r =
lim inf µ
·
r + 1 l / +∞
r+1
l / +∞
This yields (9). Since µ
b(n) > 0 for every n ≥ 0, it follows that

inf µ
b pk11 . . . pkr r > 0,
ki ≥0
1≤i≤r

and Theorem 2.1 is proved.

5.2. The case of the Furstenberg set. — Theorem 2.1 applies to the Furstenberg
′
set F = {2k 3k ; k, k′ ≥ 0} and shows the existence of a measure µ ∈ Pc (T) such that
′

inf
µ
b(2k 3k ) > 0
′

k,k ≥0

(the fact that the measure µ can be supposed to have nonnegative Fourier coefficients can
be extracted from the proof of Theorem 2.1, or deduced formally from Theorem
2.1 by
√
g
considering the measure µ ∗ µ
e). By Corollary 4.3, this means that Kaz(F ) < 2.
As mentioned in the introduction, it is natural to look for the optimal constant δ ∈ (0, 1)
for which there exists a measure µ ∈ Pc (T) such that
(10)

′

inf
µ
b(2k 3k ) ≥ δ
′

k,k ≥0

This is equivalent to asking whether F is a Kazhdan set in Z, and if yes, with which
(modified) Kazhdan constant. The best result which can be obtained via the methods

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

17

presented here is that there exists a measure µ ∈ Pc (T) satisfying (10) for every δ ∈
(0, 1/2): this is the content of Theorem 2.2, which we now prove.
Proof of Theorem 2.2. — The proof goes along the same lines as that of Theorem 2.1, but
it involves Theorem 2.6 instead of Theorem 2.4.
Fix δ ∈ (0, 1/2). There exist by Theorem 2.6 two measures µ1 , µ2 ∈ Pc (T) such that
√
′
|b
µ1 (2k 3k )| ≥ 2δ for every k ≥ 0 and every 0 ≤ k′ ≤ k
and
′

|b
µ2 (2k 3k )| ≥

√
2δ

for every k′ ≥ 0 and every 0 ≤ k ≤ k′ .

e1 + µ2 ∗ µ
e2 ) has nonnegative Fourier coefficients and satisfies
The measure µ = 12 (µ1 ∗ µ
′
µ
b(2k 3k ) ≥ δ for every k, k′ ≥ 0.
′

It then follows from Theorem 4.1 that if {2k 3kp; k, k′ ≥ 0} is a Kazhdan subset of Z,
its modified Kazhdan constant must be less than 2(1 − δ) for every δ ∈ (0, 1/2), so must
be at most 1.
That the bound 1/2 can be further improved does not seem clear at all, and we do not
know whether there exists for every δ ∈ [1/2, 1) a measure µ ∈ Pc (T) such that
′

inf
µ
b(2k 3k ) ≥ δ.
′

k,k ≥0

′

Question 5.1. — Is the Furstenberg set {2k 3k ; k, k′ ≥ 0} a Kazhdan set in Z?
Note that a lacunary semigroup {an ; n ≥ 0}, a ≥ 2, cannot be a Kazhdan set (see [5,
Ex. 5.2]).
Along the same lines, one can also ask for which values of δ ∈ (0, 1] there exists a measure
′
/ +∞. The proof of Theorem
µ ∈ Pc (T) such that lim inf µ
b(2k 3k ) ≥ δ as max(k, k′ )
2.1 allows us to exhibit a measure µ ∈ Pc (T) with nonnegative Fourier coefficients (namely
′
/ +∞. Again, we do
µ = (µ1 + µ2 )/2) such that lim inf µ
b(2k 3k ) ≥ 1/2 as max(k, k′ )
not know whether the constant 1/2 can be improved. The strongest statement which
could be expected in this direction is the existence of a measure µ ∈ Pc (T) such that
′
/ +∞. This would show that the Furstenberg sequence is
/ 1 as max(k, k ′ )
µ
b(2k 3k )
a rigidity sequence for weakly mixing dynamical systems. This natural question is raised
in Remark 3.12 (b) of [8] and we record it anew here:
Question 5.2. — Is the Furstenberg sequence a rigidity sequence for weakly mixing dynamical systems?
5.3. Examples of rigidity sequences. — Theorems 2.3 and 2.4 allow us to retrieve
directly all known examples of rigidity sequences from [8], [18], [2], [1] and [21]. The
only examples of rigidity sequences not covered by our results are those of [20]. Indeed,
Fayad and Kanigowski construct in [20] examples of rigidity sequences (nk )k≥0 such that
{λnk ; k ≥ 0} is dense in T for every λ = e2iπθ ∈ T with θ ∈ R \ Q, and there exist for every
integer p ≥ 2 infinitely many integers k such that p does not divide nk . So such sequences
never satisfy the assumption of Theorem 2.3.
We briefly list here some of the examples of rigidity sequences which can be obtained
from Theorems 2.3 and 2.4. Our first example is that of Fayad and Thouvenot in [21].

18

CATALIN BADEA & SOPHIE GRIVAUX

Example 5.3. — [21] If the sequence (nk )k≥0 is such that there exists λ = e2iπθ ∈ T,
/ +∞, (nk )k≥0 is a rigidity sequence.
/ 1 as k
with θ ∈ R \ Q, such that λnk
/ 1 with λ =
This result of [21] follows directly from Theorem 2.3. Indeed, if λnk
pn
/ 1 for every p ∈ Z. Since θ is irrational, the set {λp ; p ∈ Z} is
θ ∈ R \ Q, λ k
dense in T, and Theorem 2.3 applies.

e2iπθ ,

Example 5.4. — [8], [18] If (nk )k≥0 is a strictly increasing sequence of integers such that
nk |nk+1 for every k ≥ 0, (nk )k≥0 is a rigidity sequence.
Indeed, under the assumption of Example 5.4, the set C = {λ ∈ T ; λnk
all nk -th roots of 1, k ≥ 0, and is hence dense in T.
Theorem 2.4 shows that Example 5.4 can be improved into

/ 1} contains

Example 5.5. — Let (mk )k≥0 be a strictly increasing sequence of integers such that
/ N be a strictly increasing function. Order the
mk |mk+1 for every k ≥ 0. Let ψ : N
′
′
set {k mk ; k ≥ 0 and 1 ≤ k ≤ ψ(k)} as a strictly increasing sequence (nk )k≥0 . Then
(nk )k≥0 is a rigidity sequence.
′

/ +∞ uniformly in k ′ } contains all
/ 1 as k
Indeed, the set C ′ = {λ ∈ T ; λk mk
mk -th roots of 1, and is dense in T. So Theorem 2.4 applies.
For instance, if (rk )k≥0 is any sequence of positive integers, the sequence (nk )k≥0 obtained by ordering the set {k′ 2k ; k ≥ 0, 1 ≤ k′ ≤ rk } in a strictly increasing sequence is
a rigidity sequence. This provides new examples of rigidity sequences (nk )k≥0 such that
nk+1
/ +∞.
/ 1 as k
nk
/ +∞ as
Example 5.6. — Let (rk )k≥0 be any sequence of positive integers with rk
/
k
+∞. The sequence (nl )l≥0 obtained by ordering in a strictly increasing fashion
n
/ 1 as
the set {j2k ; k ≥ 0, 1 ≤ j ≤ rk } is a rigidity sequence which satisfies nl+1
l
/ +∞.
l

Proof. — It suffices to show that for every ε > 0 and every l sufficiently large there exists
n
l′ > l such that nll′ < 1 + ε.
– Suppose first that nl = j2k for some k ≥ 0 and some 1/ε < j < rk . Then taking
n
nl′ = (j + 1)2k , we have nll′ = j+1
j < 1 + ε.
k
′
– Suppose next that nl = j2 for some k ≥ 0 and some 1 ≤ j ≤ 1/ε. Fix an integer
p such that 2−p < ε. If l is sufficiently large, we have rk−p > 2p /ε. Set j ′ = j2p . Since
j ′ ≤ 2p /ε < rk−p , the integer nl′ = (j ′ + 1)2k−p appears in the sequence (nl )l≥0 . Also,
since nl′ = (j ′ + 1)2k−p > j2k = nl , we have l′ > l, and
nl ′
(j ′ + 1)2k−p
(j ′ + 1) −p j + 2−p
=
=
2 ≤
< 1 + 2−p < 1 + ε.
k
nl
j2
j
j
– The last case we have to deal with is when nl = rk 2k for some k ≥ 0. Let j ′ ≥ 1 be
such that j ′ ≤ rk /2 < j ′ + 1. Then j ′ < rk+1 , and if we set nl′ = (j ′ + 1)2k+1 , the integer
nl′ appears in the sequence (nl )l≥0 . We have
(j ′ + 1)2k+1
2(j ′ + 1)
2
nl ′
=
=
≤1+
<1+ε
k
nl
rk 2
rk
rk
if k is sufficiently large, and this terminates the proof.

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

19

Example 5.7. — [1] (a) Let (dk )k≥0 be a strictly increasing sequence of positive integers
of density zero. There exists a strictly increasing sequence of integers (nk )k≥0 which is a
rigidity sequence and satisfies nk ≤ dk for every k ≥ 0.
(b) Let (dk )k≥0 be a sequence of real numbers with dk ≥ k for every k ≥ 0 and
limk→+∞ dk /k = +∞. There exists a strictly increasing sequence of integers (nk )k≥0
which is a rigidity sequence and satisfies nk ≤ dk for every k ≥ 0.
This has been proved by Aaronson in [1, Th. 4]; a simpler construction with the weaker
conclusion that nk ≤ dk for infinitely many k was given in [8, Prop. 3.18]. The proof
given below uses Theorem 2.3 and a result of Bugeaud [16].
Proof. — As the statement (a) is a simple consequence of (b), we only give the proof of
(b). Set g0 = 1 and gk = dk /k for every k ≥ 1. Then (gk )k≥0 is a sequence of reals with
gk ≥ 1 for every k ≥ 0 which tends to infinity (notice that for (a) this holds since (dk )k≥0
is a sequence of density zero). Using (a particular case of) [16, Th. 1], we obtain that
there exists for every fixed irrational number θ an increasing sequence (nk )k≥0 of positive
/ 1. It follows from
integers such that nk ≤ kgk = dk for every k ≥ 1 and exp(2iπθ)nk
Example 5.3 that (nk )k≥0 is a rigidity sequence.
Example 5.8. — Let (mk )k≥0 be a strictly increasing sequence of positive integers with
/ +∞. There exists a strictly increasing sequence of integers (nk )k≥0 which
mk+1 − mk
is a rigidity sequence and satisfies mk ≤ nk < mk+1 for every k ≥ 0.
Proof. — The proof is exactly the same as the preceding one, replacing the result from
[16] by [9, Obs. 1.36].
5.4. Exceptional sets for (almost) uniform distribution. — Let (nk )k≥0 be a
strictly increasing sequence of integers, and let ν ∈ M(T) be a (finite) complex Borel
measure on T. We stress that ν is not necessarily a probability measure. Given θ ∈ R,
the sequence (nk θ)k≥0 is said ([30], [28, p. 53]) to be almost uniformly distributed with
respect to ν if there exists a strictly increasing sequence (Nj )j≥1 of positive integers such
that for every arc I ⊂ T whose endpoints are not atoms (mass-points) for ν one has
1
# {n ≤ Nj : exp(2iπnk θ) ∈ I} = ν(I).
lim
j→+∞ Nj
The analog of Weyl’s criterion states that (nk θ)k≥0 is almost uniformly distributed with
respect to ν if and only if there exists a strictly increasing sequence (Nj )j≥1 of positive
integers such that
Nj
1 X
lim
exp(m2iπnk θ) exists
j→∞ Nj
k=1

for every m ∈ Z.

In this case, the limit is ν̂(m). It can also be proved that (nk θ)k≥0 is almost uniformly
distributed with respect to ν if and only if there exists a strictly increasing sequence
(Nj )j≥1 of positive integers such that
Nj

1 X
f e2iπnk θ
Nj
k=1

/

Z

f dµ
T

as

j

/ +∞

for every f ∈ C(T).

We now denote by W ((nk )k≥0 , ν), the exceptional set of almost uniform distribution of
(nk ) with respect to ν. This is the set of all θ ∈ R such that (nk θ)k≥0 is not almost

CATALIN BADEA & SOPHIE GRIVAUX

20

uniformly distributed with respect to ν. We will write U ((nk )k≥0 , ν) for the exceptional
set of (classical) uniform distribution of (nk ) with respect to ν, which corresponds to the
case where Nj = j for every j ≥ 1.
The size of the exceptional set U ((nk )k≥0 , ν) has been studied in many works, in particular in the case where ν is the normalized Lebesgue measure on T. In this case, we write
it as U ((nk )k≥0 ). If the sequence (nk )k≥0 is lacunary, U ((nk )k≥0 ) is uncountable, and
even of Hausdorff dimension 1 ([19], see also [24]). See also [34] and [32] for a stronger
result. On the other hand, it is known (see [11], [13]) that among various natural classes
of random sequences of integers, almost all sequences (nk )k≥0 satisfy U ((nk )k≥0 ) = Q.
/ 1 as
These typical random sequences (nk )k≥0 are sublacunary, i.e. satisfy nk+1 /nk
/ +∞. Nonetheless, examples of sublacunary sequences (nk )k≥0 with U ((nk )k≥0 ) unk
countable were constructed in [19] (see also [6]). Concerning the size of W ((nk )k≥0 , ν) we
refer for instance to [33], [24] and [27]. See also [15] for other references.
Our results about the size of W ((nk )k≥0 , ν) rely on the following generalization of Proposition 4.6, which provides a link between the size of the exceptional set W ((nk )k≥0 , ν) and
the modified Kazhdan constant of the set {nk ; k ≥ 0}.
Proposition 5.9. — Let (nk )k≥0 be a strictly increasing sequence of positive integers with
n0 = 1, and let ν ∈ M(T) with ν 6= δ{1} . If W ((nk )k≥0 , ν) is finite or countable infinite,
Q = {nk ; k ≥ 0} is a Kazhdan subset of Z, and
p
g
Kaz(Q)
≥ 2(1 − ℜe νb(1)).

Proof. — Fix γ ∈ (0, 1 − ℜe νb(1)), and let µ be a probability measure on T such that
supk≥0 (1 − ℜe µ
b(nk )) < γ. Then
1 − ℜe

Z  X
N

1
λnk dµ(λ) < γ
T N
k=1

for every N ≥ 1.

Suppose that the measure µ is continuous. Since there exists a strictly increasing sequence
(Nj )j≥1 of integers such that
Nj
1 X nk
λ
Nj
k=1

/ν
b(1)

as j

/ +∞

for every λ ∈ T \ C,

where C is a finite or countable infinite subset of T, we have 1 − ℜe νb(1) ≤ γ, which
contradicts our initial assumption. So µ has a discrete part.
p It then follows from Theorem
4.1 that the modified Kazhdan constant of Q is at least 2(1 − ℜe νb(1)).

The following result provides an example of a nonlacunary semigroup (nk )k≥0 whose
associated exceptional sets W ((nk )k≥0 , ν) with respect to ν are uncountable for a large
class of measures ν ∈ M(T).
Theorem 5.10. — Denote by (nk )k≥0 the sequence obtained by ordering the Furstenberg
′
set F = {2k 3k ; k, k ′ ≥ 0} in a strictly increasing fashion. For every measure ν ∈ M(T)
such that ℜe νb(1) < 1/2, the set W ((nk )k≥0 , ν) is uncountable.

Proof of Theorem 5.10. — Fix ν ∈ M(T), and suppose that U ((nk )k≥0 , ν) is at most
g ) ≤ 1 by Theorem 2.2, it follows from Proposition 5.9 that
countable.
Since Kaz(F
p
2(1 − ℜe νb(1)) ≤ 1, i.e. that ℜe νb(1) ≥ 1/2. This proves Theorem 5.10.

CONTINUOUS PROBABILITY MEASURES WITH LARGE FOURIER COEFFICIENTS

21

References
[1] J. Aaronson, Rational ergodicity, bounded rational ergodicity and some continuous measures
on the circle, Israel J. Math. 33 (1979), 181–197.
[2] J. Aaronson, M. Hosseini, M. Lemańczyk, IP-rigidity and eigenvalue groups, Erg. Th.
Dynam. Syst. 34 (2014), 1057–1076.
[3] T. Adams, Tower multiplexing and slow weak mixing, Colloq. Math. 138 (2015), 47–72.
[4] C. Badea, S. Grivaux, Kazhdan sets in groups and equidistribution properties, J. Funct.
Anal. 273 (2017), 1931–1969.
[5] C. Badea, S. Grivaux, Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group Z (Actes du 1er congrès national de la SMF - Tours 2016),
Séminaires et Congrès 31, Société Mathématique de France (2017).
[6] R.C. Baker, On a theorem of Erdös and Taylor, Bull. London Math. Soc. 4 (1972), 373–374.
[7] B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s Property (T), New Mathematical
Monographs 11, Cambridge University Press (2008).
[8] V. Bergelson, A. del Junco, M. Lemańczyk, J. Rosenblatt, Rigidity and nonrecurrence along sequences, Erg. Th. Dynam. Syst. 34 (2014), 1464–1502.
[9] V. Bergelson, D. Simmons, New examples of complete sets, with connections to a Diophantine theorem of Furstenberg, Acta Arith. 177 (2017), 101–131.
[10] M. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math.
Soc. 122 (1994), 67–70.
[11] M. Boshernitzan, Density modulo 1 of dilations of sublacunary sequences, Adv. Math. 108,
(1994) 104–117.
[12] M. Boshernitzan, Homogeneously distributed sequences and Poincaré sequences of integers
of sublacunary growth, Monatsh. Math. 96 (1983), 173–181.
[13] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J.
Math. 61 (1988), 39–72.
[14] J. Bourgain, E. Lindenstrauss, P. Michel, A. Venkatesh, Some effective results for
×a × b, Erg. Th. Dynam. Syst. 29 (2009), 1705–1722.
[15] Y. Bugeaud, Distribution modulo one and Diophantine approximation, Cambridge Tracts
in Mathematics 193, Cambridge University Press (2012).
[16] Y. Bugeaud, On sequences (an ξ)n≥1 converging modulo 1, Proc. Amer. Math. Soc. 137
(2009), 2609–2612.
[17] I. Chatterji, D. Witte Morris, R. Shah, Relative property (T) for nilpotent subgroups,
preprint 2017, arXiv:1702.01801.
[18] T. Eisner, S. Grivaux, Hilbertian Jamison sequences and rigid dynamical systems, J. Funct.
Anal. 261 (2011), 2013–2052.
[19] P. Erdös, S. Taylor, On the set of points of convergence of a lacunary trigonometric series
and the equidistribution properties of related sequences, Proc. London Math. Soc. 7 (1957),
598–615.
[20] B. Fayad, A. Kanigowski, Rigidity times for a weakly mixing dynamical system which are
not rigidity times for any irrational rotation, Erg. Th. Dynam. Syst. 35 (2015), 2529–2534.
[21] B. Fayad, J.-P. Thouvenot, On the convergence to 0 of mn ξ mod 1, Acta Arith. 165
(2014), 327–332.
[22] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine
approximation, Math. Systems Theory 1 (1967), 1–49.
[23] S. Grivaux, IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products, Studia Math. 215 (2013), 237–259.
[24] H. Helson, J.-P. Kahane, A Fourier method in Diophantine problems, J. Analyse Math.
15 (1965), 245–262.

22

CATALIN BADEA & SOPHIE GRIVAUX

[25] M. Hochman, Geometric rigidity of ×m invariant measures, J. of the European Math. Soc.
14 (2012), 1539–1563.
[26] M. Hochman, Lectures on dynamics, fractal geometry, and metric number theory, J. of Mod.
Dyn. 8 (2014), 437–497.
[27] J.-P. Kahane, Sur les mauvaises répartitions modulo 1, Ann. Inst. Fourier (Grenoble) 14
(1964), 519–526.
[28] L. Kuipers, H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience (1974).
[29] E. Lindenstrauss, Rigidity of multiparameter actions, Probability in mathematics, Israel
J. Math. 149 (2005), 199–226.
[30] R. Lyons, Fourier-Stieltjes coefficients and asymptotic distribution modulo 1, Ann. of Math.
122 (1985), 155–170.
[31] R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 (1988), 219–224.
[32] B. de Mathan, Sur un problème de densité modulo 1, C. R. Acad. Sci. Paris 287 (1978),
277–279.
[33] I. Piatetski-Shapiro, On the laws of distribution of the fractional parts of an exponential
function, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 47–52 (in Russian).
[34] A. Pollington, On the density of sequence (nk ξ), Illinois J. Math. 23 (1979), 511–515.
[35] D. Rudolph, ×2 and ×3 invariant measures and entropy, Erg. Th. Dynam. Syst. 10 (1990),
395–406.
[36] A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. 45 (2008), 117–134.

Catalin Badea, Laboratoire Paul Painlevé, UMR 8524, Université de Lille, Cité Scientifique, Bâtiment
M2, 59655 Villeneuve d’Ascq Cedex, France • E-mail : catalin.badea@univ-lille1.fr
Sophie Grivaux, CNRS, Laboratoire Paul Painlevé, UMR 8524, Université de Lille,
Cité Scientifique, Bâtiment M2, 59655 Villeneuve d’Ascq Cedex, France
E-mail : sophie.grivaux@math.univ-lille1.fr

