On generalized lacunary series

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 $c_k=0$.