Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Badea, Cătălin; Grivaux, Sophie

doi:10.4171/cmh/482

Cited by 6 publications

(28 citation statements)

References 64 publications

(199 reference statements)

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“…Then there is a rigid sequence (r n ) such that r n ≤ d n for all n ∈ N [1]. See [5], [10], [4] and [11] for more exhaustive lists of rigid sequences.…”

Section: Definitionmentioning

confidence: 99%

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Nilsequences and multiple correlations along subsequences

Le¹

2018

Ergod. Th. Dynam. Sys.

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The results of Bergelson-Host-Kra and Leibman say that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a null sequence (a sequence that goes to zero in density). We refine their results by proving that the null sequence goes to zero in density along polynomials evaluated at primes and Hardy sequence (⌊n c ⌋). On the other hand, given a rigid sequence, we construct an example of correlation whose null sequence does not approach zero in density along that rigid sequence. As a corollary of a lemma in the proof, the formula for the pointwise ergodic average along polynomials of primes in a nilsystem is also obtained.

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“…Then there is a rigid sequence (r n ) such that r n ≤ d n for all n ∈ N [1]. See [5], [10], [4] and [11] for more exhaustive lists of rigid sequences.…”

Section: Definitionmentioning

confidence: 99%

“…Remark. In a recent paper, Badea and Grivaux [4] show that there exists a continuous measure σ on T such that lim inf n,m∈Nσ (2 n 3 m ) > 0. Hence sequence (2 n 3 m ) when ordering in the increasing fashion also satisfies Proposition 6.1 hypothesis.…”

mentioning

confidence: 99%

Nilsequences and multiple correlations along subsequences

Le¹

2018

Ergod. Th. Dynam. Sys.

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“…z is not a root of 1) such that z n k → 1 as k → ∞, then (n k ) k≥0 is a rigidity sequence; and a simpler proof of this result was found by Fayad and Thouvenot in [20]. The result was further generalized in [5], and this was applied to the resolution of a conjecture of Lyons [38] related to Furstenberg's ×2 -×3 conjecture. On the other hand, examples of rigidity sequences (n k ) with the property that the set {z n k ; k ≥ 0} is dense in T for every irrational z ∈ T were constructed in [19].…”

Section: Introductionmentioning

confidence: 89%

“…This leads to an equidistribution criterion implying that a set is Kazhdan, as well as to explicit characterizations of Kazhdan sets in many groups without Property (T), such as locally compact abelian groups or Heisenberg groups. See also [11] for examples of Kazhdan sets in other Lie groups, and [5] for dynamical applications.…”

Section: Introductionmentioning

confidence: 99%

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Rigidity sequences, Kazhdan sets and group topologies on the integers

Badea

Grivaux

Matheron

2021

JAMA

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We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in Z for the Bohr topology, a result originally due to Griesmer.

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Rigidity, weak mixing, and recurrence in abelian groups

Ackelsberg¹

2022

DCDS

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The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions extend to this setting:1. If <inline-formula><tex-math id="M2">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.2. There exists a sequence <inline-formula><tex-math id="M3">\begin{document}$ (r_n) $\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.The first of these results was shown for <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type="bibr" rid="b1">1</xref>], Fayad and Thouvenot [<xref ref-type="bibr" rid="b20">20</xref>], and Badea and Grivaux [<xref ref-type="bibr" rid="b2">2</xref>]. The latter was established in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type="bibr" rid="b23">23</xref>]. While techniques for handling <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>, while others exhibit new phenomena.

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Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

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 k 1 ≥0,...,kr ≥0

Cited by 6 publications

References 64 publications

Nilsequences and multiple correlations along subsequences

Nilsequences and multiple correlations along subsequences

Rigidity sequences, Kazhdan sets and group topologies on the integers

Rigidity, weak mixing, and recurrence in abelian groups

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