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

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 $$\underset{k_1\ge 0,\dots ,k_r\ge 0}{\inf }\mathrm{\mid }\widehat{\mu }(p_1^{k_1}\dots p_r^{k_r})\mathrm{\mid }>0.$$ This results applies in particular to the Furstenberg set $F=\{2^k{3}^{k^{\mathrm{\prime }}};k\ge 0,k^{\mathrm{\prime }}\ge 0\}$, 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.