On the convergence to 0 of mₙξmod 1

• Autorzy: Bassam Fayad , Jean-Paul Thouvenot
• Języki publikacji: EN

Abstrakty

EN

We show that for any irrational number α and a sequence ${m_l}_{l∈ℕ}$ of integers such that $lim_{l→∞} |||m_l α||| = 0$, there exists a continuous measure μ on the circle such that $lim_{l→∞} ∫_{𝕋} |||m_l θ||| dμ(θ) = 0$. 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 ${m_l}_{l∈ℕ}$ of integers such that $|||m_l α||| → 0$ and such that $m_l θ[1]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

Kategorie tematyczne

• 11K06: General theory of distribution modulo 1
• 37A30: Ergodic theorems, spectral theory, Markov operators
• 11J71: Distribution modulo one
• 11K31: Special sequences

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 165
• Numer: 4
• Strony: 327-332

Daty

wydano

2014

Twórcy

autor

Bassam Fayad

• IMJ-PRG, CNRS UMR 7586, UP7D-Campus Grand Moulin, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France

autor

Jean-Paul Thouvenot

• LPMA, Université Pierre et Marie Curie, 4 pl. Jussieu, 75252 Paris, France

Identyfikatory

DOI

10.4064/aa165-4-2