# Lacunary sequence

$\newcommand{\seq}{\left(#1\right)} % \newcommand{\seq}{\left\{#1\right\}}$
A sequence of numbers $\seq{n_k}$ such that $n_{k+1} / n_k \geq \lambda > 1$; this class of sequences is denoted by $\Lambda$ and is used, in particular, in the theory of lacunary series and in the theory of lacunary trigonometric series. There are generalizations of the class $\Lambda$. For example, the class $B_2$: $\seq{n_k} \in B_2$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm n_{k_2}] = m$ (where $n_{k_1} > n_{k_2}$ and $[a]$ is the integer part of the number $a$) does not exceed $A$ for any integer $m$; the class $R$: $\seq{n_k} \in R$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm \cdots \pm n_{k_p}] = m$ (where $n_{k_1} > \cdots > n_{k_p}$) does not exceed $A^p$ for any $p=2,3,\ldots$ and any integer $m$; and the classes $\Lambda_\sigma$, $B_{2\sigma}$, $R_\sigma$, consisting of sequences that split into finitely-many sequences of the classes $\Lambda$, $B_2$, $R$, respectively.