Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation

Abstract

We construct an increasing sequence of natural numbers $(m_{n})_{n=1}^{+\infty }$ with the property that $(m_{n}{\it\theta}[1])_{n\geq 1}$ is dense in $\mathbb{T}$ for any ${\it\theta}\in \mathbb{R}\setminus \mathbb{Q}$, and a continuous measure on the circle ${\it\mu}$ such that $\lim _{n\rightarrow +\infty }\int _{\mathbb{T}}\Vert m_{n}{\it\theta}\Vert \,d{\it\mu}({\it\theta})=0$. Moreover, for every fixed $k\in \mathbb{N}$, the set $\{n\in \mathbb{N}:k\nmid m_{n}\}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.