On generalized lacunary series

TL;DR

This paper studies the convergence properties of generalized lacunary series formed by sums of lacunary sequences, establishing almost everywhere convergence under certain conditions and deriving new inequalities related to these series.

Contribution

It introduces a new class of generalized lacunary series, proves almost everywhere convergence for these series under specific conditions, and establishes a Khintchine type inequality with sharp growth rates.

Findings

Series converge a.e. if $orall ext{lacunary sequences with } rac{n_{k+1}}{n_k}> ext{threshold}$

New Khintchine type inequality for sums of generalized lacunary series

Sharp growth rate of constants in inequalities as $p o fty$

Abstract

Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>\lambda>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$. We prove if $\lambda>\lambda_l$, then any series \begin{equation} \sum_kc_ke^{im_kx},\qquad (1) \end{equation} with $\sum_k|c_k|^2<\infty$ converges almost everywhere after any rearrangement of the terms, where $1<\lambda_l<2$ is a certain critical value. We establish this property, proving a new Khintchine type inequality $\|S\|_p\le C_{l,\lambda,p}\|S\|_2$, $p>2$, where $S$ is a finite sum of form (1). For $\lambda\ge 3$, we also establish a sharp rate $p^{l/2}$ for the growth of the constant $C_{l,\lambda,p}$ as $p\to\infty$. Such an estimate for the Rademacher chaos sums was proved independently by Bonami and Kiener. In the case of $\lambda\ge 3$ we also establish some inverse convergence properties of series (1): 1) if series (1) converges a.e., then $\sum_k|c_k|^2<\infty$, 2) if it a.e. converges to zero, then $c_k=0$.