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.

中文翻译：

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Kazhdan 常数，具有大傅立叶系数和刚性序列的连续概率测度

利用 Fayad 和 Thouvenot 为弱混合动力系统构建的刚性序列，我们表明对于每个整数 $p_{1},\dots,p_{r}$ 都存在单位圆上的连续概率测度 $\mu $ $\mathbb{T}$ 使得 \[ \inf_{k_{1}\ge 0,\dots,k_{r}\ge 0}|\widehat{\mu }(p_{1}^{k_{1 }}\dots p_{r}^{k_{r}})|>0。\] 这个结果特别适用于 Furstenberg 集合 $F=\{2^{k}3^{k'}\,;\,k\ge 0,\ k'\ge 0\}$，并且反驳了一个1988 年里昂猜想的灵感来自 Furstenberg 著名的 $\times 2$-$\times 3$ 猜想。我们还估计了 $F$ 的修正 Kazhdan 常数，并获得了刚性序列的一般结果，这使我们能够基本上检索此类序列的所有已知示例。