Maximal gaps for dilated lacunary integer sequences

Source abstract

Abstract Let $(a_{n})_{n\ge 1}\subset \mathbb N$ be a lacunary sequence, $a_{n+1}\ge q a_{n}$ for $q>1$. For $x\in \mathbb T$, we study the maximal gap $G_{N}(x)$ of the finite orbit $\{a_{1}x,\ldots ,a_{N}x\}$. We prove that, for Lebesgue-almost every $x$, $$\begin{align*} & \frac12 \le \liminf_{N\to\infty}\frac{NG_{N}(x)}{\log N} \le \limsup_{N\to\infty}\frac{NG_{N}(x)}{\log N} \le \frac{q+1}{q-1}\,. \end{align*}$$ If, in addition, $a_{n}\mid a_{n+1}$ for every $n$, then this can be improved to $$\begin{align*} & \lim_{N\to\infty}\frac{NG_{N}(x)}{\log N}=1 \end{align*}$$ for Lebesgue-almost every $x$.