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

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