On the convergence to 0 of mₙξmod 1

Bassam Fayad, and Jean-Paul Thouvenot. "On the convergence to 0 of mₙξmod 1." Acta Arithmetica 165.4 (2014): 327-332. <http://eudml.org/doc/279765>.

@article{BassamFayad2014,
abstract = {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 $lim_\{l→∞\} ∫_\{\} |||m_l θ||| dμ(θ) = 0$. 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.},
author = {Bassam Fayad, Jean-Paul Thouvenot},
journal = {Acta Arithmetica},
keywords = {limit points of Kronecker sequences; weak mixing; rigidity sequences; exceptional points},
language = {eng},
number = {4},
pages = {327-332},
title = {On the convergence to 0 of mₙξmod 1},
url = {http://eudml.org/doc/279765},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Bassam Fayad
AU - Jean-Paul Thouvenot
TI - On the convergence to 0 of mₙξmod 1
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 4
SP - 327
EP - 332
AB - 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 $lim_{l→∞} ∫_{} |||m_l θ||| dμ(θ) = 0$. 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.
LA - eng
KW - limit points of Kronecker sequences; weak mixing; rigidity sequences; exceptional points
UR - http://eudml.org/doc/279765
ER -