We construct an increasing sequence of natural numbers $(m_n)_{ n=1}^{ +\infty}$ with the property that $(m_n \th [1])_{ n\geq 1}$ is dense in $\T$ for any $\th \in \R\setminus \Q$, and a continuous measure on the circle $\mu$ such that $\lim_{ n\to +\infty}\int_{ \T}\|m_n\theta\|d\mu(\theta)=0$. Moreover, for every fixed $k\in \N$, the set $\{ n\in \N:\,k\nmid m_n \}$ 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.
Categories: math.DS