On the convergence to $0$ of $m_n\xi$ mod $1$

Abstract

We show that for any irrational number $\a$ and a sequence of integers $\{m_l{\}}_{l\in \mathbb{N}}$ such that $\displaystyle{\lim_{l\to \infty} \norm{m_l \a} = 0}$, there exists a continuous measure $\mu$ on the circle such that $\displaystyle{\lim_{l\to \infty} \int_\T \norm{m_l þ} dμ(þ) = 0}$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any $\a \in \R - \Q$, there exists a sequence of integers $\{m_l{\}}_{l\in \mathbb{N}}$ such that $\norm{m_l \a} \to 0$ and $m_l\theta [1]$ is dense on the circle if and only if $þ\notin \Q \a+\Q$.