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

Catalin Badea (1) , Sophie Grivaux (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 CNRS - Centre National de la Recherche Scientifique

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.

Domaines: Mathématiques [math]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

Catalin Badea, Sophie Grivaux. Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences. Commentarii Mathematici Helvetici, 2019, 95 (1), pp.99-127. ⟨10.4171/cmh/482⟩. ⟨hal-02140446⟩