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 | µ(p k 1 1. .. p kr r)| > 0. This results applies in particular to the Furstenberg set F = {2 k 3 k ; 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.