  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.
