Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation

Abstract

We construct an increasing sequence of natural numbers (m n ) +∞ n=1 with the property that (m n θ[1]) n 1 is dense in T for any θ ∈ R \ Q, and a continuous measure on the circle µ such that lim n→+∞ T m n θ dµ(θ) = 0. Moreover, for every fixed k ∈ N, the set {n ∈ N : k ∤ m n } is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.

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

“…Remark also that the assumption of Theorem 2.4 is in particular satisfied if the set [21] and [24]). We state separately as Corollaries 2.5 and 2.6 the parts of Theorems 2.3 and 2.4 dealing with rigidity sequences: Corollary 2.5.…”

“…The only examples of rigidity sequences not covered by our results are those of [21] and [24]. Indeed, Fayad and Kanigowski construct in [21] examples of rigidity sequences (n k ) k≥0 such that {λ n k ; k ≥ 0} is dense in T for every λ = e 2iπθ ∈ T with θ ∈ R \ Q, and there exist for every integer p ≥ 2 infinitely many integers k such that p does not divide n k . So such sequences never satisfy the assumption of Corollary 2.5.…”

Rigidity sequences, Kazhdan sets and group topologies on the integers

“…Rigidity sequences are studied in detail in several papers, among which we mention [8,14,2,20,19,29]. All examples of non-Kazhdan sequences given above can be seen to be rigidity sequences.…”

“…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]. Furthermore, Griesmer [29] proved that there exist rigidity sequences (n k ) with the property that every translate R of the set {n k ; k ≥ 0} is a set of recurrence (in the terminology of [23], a Poincaré set), which means that for any measurepreserving system (X, B, m; T ) and every A ∈ B with m(A) > 0, there exists r ∈ R \ {0} such that m(A ∩ T −r A) > 0.…”

Recurrence, rigidity, and popular differences

“…[6],Theorem 2). There is a set of rigidity S ⊆ Z such that for all irrational α, {nα mod 1 : n ∈ S} is dense in R/Z, and for all rational α, {nα mod 1 : n ∈ S} = {nα mod 1 : n ∈ Z}.In light of Proposition 5.2 and Remark 5.1, Theorem 2.1 implies Theorem 5.3 and the following strengthening.…”