On the convergence to $0$ of $m_n \xi\, {\rm mod}\,1$
Volume 165 / 2014
Acta Arithmetica 165 (2014), 327-332
MSC: 11K31, 37A30; Secondary 11J71, 11K06.
DOI: 10.4064/aa165-4-2
Abstract
We show that for any irrational number $\alpha$ and a sequence $\{m_l\}_{l\in \mathbb N}$ of integers such that $\lim_{l\to \infty} |\!|\!|m_l \alpha |\!|\!| = 0$, there exists a continuous measure $\mu$ on the circle such that $ \lim_{l\to \infty} \int_{\mathbb T} |\!|\!|m_l \theta |\!|\!| \,d\mu(\theta) = 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 $\alpha \in \mathbb R - \mathbb Q$, there exists a sequence $\{m_l\}_{l\in \mathbb N}$ of integers such that $|\!|\!|m_l \alpha|\!|\!|\to 0$ and such that $m_l \theta [1]$ is dense on the circle if and only if $\theta \notin \mathbb Q \alpha+\mathbb Q$.
