Lacunary sequences in analysis, probability and number theory

TL;DR

This paper reviews the theory of lacunary sequences across analysis, probability, and number theory, highlighting recent advances and establishing new criteria for the central limit theorem in subsequences.

Contribution

It introduces new necessary and sufficient conditions for subsequences to satisfy the strong central limit theorem, extending previous resonance theorems.

Findings

Characterization of sequences allowing strong CLT in subsequences

Connections established between lacunary sums and number theory topics

New results on conditions for convergence in probability and distribution

Abstract

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.