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We show that for any irrational number α and a sequence of integers such that , there exists a continuous measure μ on the circle such that . 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 of integers such that and such that is dense on the circle if and only if θ ∉ ℚα + ℚ.
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 -