On the convergence to 0 of mₙξmod 1

Bassam Fayad; Jean-Paul Thouvenot

Acta Arithmetica, Tome 166 (2014), p. 327-332 / Harvested from The Polish Digital Mathematics Library

Résumé

We show that for any irrational number α and a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $l i m_{l \to \infty} | | | m_{l} α | | | = 0$ , there exists a continuous measure μ on the circle such that $l i m_{l \to \infty} \int_{} | | | m_{l} θ | | | d μ (θ) = 0$ . 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 α ∈ ℝ - ℚ, there exists a sequence ${m_{l}}_{l \in ℕ}$ of integers such that $| | | m_{l} α | | | \to 0$ and such that $m_{l} θ [1]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

Publié le : 2014-01-01

Zbl 1310.11079

EUDML-ID : urn:eudml:doc:279765

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     author = {Bassam Fayad and Jean-Paul Thouvenot},
     title = {On the convergence to 0 of mnxmod 1},
     journal = {Acta Arithmetica},
     volume = {166},
     year = {2014},
     pages = {327-332},
     zbl = {1310.11079},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-4-2}
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Bassam Fayad; Jean-Paul Thouvenot. On the convergence to 0 of mₙξmod 1. Acta Arithmetica, Tome 166 (2014) pp. 327-332. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-4-2/