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

We construct an increasing sequence of natural numbers $(m_n)_{n=1}^{+\infty}$ with the property that $(m_n þ[1])_{n\geq 1}$ is dense in $\T$ for any $þ\in \R\setminus \Q$, and a continuous measure on the circle $μ$ such that $\lim_{n\to +\infty}\int_{\T}\|m_nθ\|dμ(θ)=0$. 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.

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