Sublacunary sequences that are strong sweeping out

2022/10/28 by Sovanlal Mondal, Mondal, Sovanlal, Madhumita Roy +3 · 1 citation
Computer Science · Mathematics · #37A30 (Primary) #37A44 #37A46 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2210.15894

openalex publication_date 2022/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An increasing sequence (a_n) of positive integers which satisfies \fraca_n+1a_n>1+η for some positive η is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out which means that in every aperiodic dynamical system we can find a set E of arbitrary small measure so that \limsup_N(1)/(N) ∑_{n≤ N}\mathbb1_E(Tⁿx)=1 and \liminf_N(1)/(N) ∑_{n≤ N}\mathbb1_E(Tⁿx)=0 almost everywhere. In this paper we improve this result by showing that if (a_n) satisfies only \fraca_n+1a_n>1+\frac1(loglog n)^1-η for some positive η then it is already strong sweeping out.

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