Maximal Gaps for Dilated Lacunary Integer Sequences

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 empty circular gap $G_N(x)$ of the finite orbit $\{a_{1}x,\dots ,a_{N}x\}$. We prove that, for Lebesgue-almost every $x$, $$\frac{1}{2}\le \underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{N{G}_N(x)}{\log N}\le \underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{N{G}_N(x)}{\log N}\le \frac{q+1}{q-1}\mathrm{.}$$ If, in addition, $a_n\mid a_{n+1}$ for every $n$, then this can be improved to $$\underset{N\to \mathrm{\infty }}{\lim }\frac{N{G}_N(x)}{\log N}=1$$ for Lebesgue-almost every $x$.