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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.

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