Maximal Gaps for Dilated Lacunary Integer Sequences

Let $(a_n)_{n\ge1}\subset\mathbb{N}$ be a lacunary sequence, $a_{n+1}\ge q a_n$ for $q>1$. For $x\in\mathbb{T}$, we study the maximal empty circular gap $G_N(x)$ of the finite orbit $\{a_1x,\ldots,a_Nx\}$. We prove that, for Lebesgue-almost every $x$, $$\frac{1}{2} \le \liminf_{N\to\infty}\frac{NG_N(x)}{\log N} \le \limsup_{N\to\infty}\frac{NG_N(x)}{\log N} \le \frac{q+1}{q-1}\,.$$ If, in addition, $a_n\mid a_{n+1}$ for every $n$, then this can be improved to $$\lim_{N\to\infty}\frac{NG_N(x)}{\log N}=1$$ for Lebesgue-almost every $x$.