On the convergence to \(0\) of \(m_n \xi \mod 1\) (Q2921805)

The article focuses on the study of sequences of the form \(\{m_l\alpha\}_{l\in \mathbb{N}}\) with \(\alpha\) being an irrational number and \(\{m_l\}_{l\in \mathbb{N}}\) a sequence of integers such that \( \|{m_l \alpha}\| \to 0\) as \(l\to \infty\) (for any \(r\in\mathbb{R}\), \(\|r \|\) denotes its distance to integers). For any such sequence \(\{m_l\alpha\}_{l\in \mathbb{N}}\), it is proved that there exists a non-atomic measure \(\mu\) on \(\mathbb{T}:=\mathbb{R}/\mathbb{Z}\) such that \(\int_\mathbb{T} \|{m_l \theta}\| d\mu(\theta) \to 0\) as \({l\to \infty}\). A direct consequence of this result is that any rigidity sequence of any ergodic transformation (on a probability space without atoms) with discrete spectrum is a rigidity sequence for some weakly mixing dynamical system. This provides a positive answer to a question posed in [\textit{V. Bergelson} et al., Ergodic Theory Dyn. Syst. 34, No. 5, 1464--1502 (2014; Zbl 1351.37002)] and has been recently proved with different methods also by \textit{T. M. Adams} [``Tower multiplexing and slow weak mixing'', Preprint, \url{arXiv:1301.0791}]. Furthermore, the authors show that for any irrational number there exists a sequence of integers for which the set of exceptional points is countable. More precisely, for any \(\alpha \in \mathbb{R}\setminus\mathbb{Q}\) they construct an increasing sequence of integers \(\{m_l\}_{l\in \mathbb{N}}\) with \( \|{m_l \alpha}\| \to 0\) as \(l\to \infty\) such that \(\{m_l\theta[1]\}_{\l\in\mathbb{N}}\) is dense in \(\mathbb{T}\) if and only if \(\theta \notin \mathbb{Q} \alpha+\mathbb{Q}\) (here \(\theta[1]\) denotes the fractional part of \(\theta\)).