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

TL;DR

This paper constructs continuous measures on the unit circle with large Fourier coefficients at specific exponential sequences, disproving a long-standing conjecture and providing new insights into rigidity sequences in dynamical systems.

Contribution

It introduces a novel construction of measures with large Fourier coefficients on exponential sequences, challenging a 1988 conjecture and advancing the understanding of rigidity in dynamical systems.

Findings

Existence of continuous measures with large Fourier coefficients on exponential sequences

Disproof of Lyons' 1988 conjecture related to Furstenberg's imes 2- imes 3 problem

Estimation of Kazhdan constants and general results on rigidity sequences

Abstract

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\dots,p_{r}$ there exists a continuous probability measure $\mu $ on the unit circle $\mathbb{T}$ such that \[ \inf_{k_{1}\ge 0,\dots,k_{r}\ge 0}|\widehat{\mu }(p_{1}^{k_{1}}\dots p_{r}^{k_{r}})|>0. \] This results applies in particular to the Furstenberg set $F=\{2^{k}3^{k'}\,;\,k\ge 0,\ k'\ge 0\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\times 2$-$\times 3$ 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.