On the convergence to $0$ of $m_n \xi\, {\rm mod}\,1$

Bassam Fayad¹ ; Jean-Paul Thouvenot²

¹IMJ-PRG, CNRS UMR 7586 UP7D–Campus Grand Moulin Bâtiment Sophie Germain, Case 7012 75205 Paris Cedex 13, France
²LPMA Université Pierre et Marie Curie 4 pl. Jussieu 75252 Paris, France

Acta Arithmetica, Tome 165 (2014) no. 4, pp. 327-332

Résumé

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 $\theta \notin \mathbb Q \alpha+\mathbb Q$.

Détail

DOI : 10.4064/aa165-4-2

Keywords: irrational number alpha sequence mathbb integers lim infty alpha there exists continuous measure circle lim infty int mathbb theta theta implies rigidity sequence ergodic transformation rigidity sequence weakly mixing dynamical system other alpha mathbb mathbb there exists sequence mathbb integers alpha theta dense circle only theta notin mathbb alpha mathbb