Sublacunary sequences that are strong sweeping out

An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+\eta$ for some positive $\eta$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out which means that in every aperiodic dynamical system we can find a set $E$ of arbitrary small measure so that ${\mathrm{lim\hspace{0.17em}sup}}_N\frac{1}{N}{\sum }_{n\le N}{\mathbb{1}}_E(T^{n}x)=1$ and ${\mathrm{lim\hspace{0.17em}inf}}_N\frac{1}{N}{\sum }_{n\le N}{\mathbb{1}}_E(T^{n}x)=0$ almost everywhere. In this paper we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac{1}{(\log \log n{)}^{1-\eta }}$ for some positive $\eta$ then it is already strong sweeping out.