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

  • Catalin Badea

    Université de Lille, Villeneuve-d'Ascq, France
  • Sophie Grivaux

    Université de Lille, Villeneuve-d'Ascq, France
Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences cover

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Abstract

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers there exists a continuous probability measure on the unit circle such that

This results applies in particular to the Furstenberg set , and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous conjecture. We also estimate the modified Kazhdan constant of and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.

Cite this article

Catalin Badea, Sophie Grivaux, Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences. Comment. Math. Helv. 95 (2020), no. 1, pp. 99–127

DOI 10.4171/CMH/482