Sublacunary sequences that are strong sweeping out

Abstract: An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}&gt;1+\eta$ 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 $$\limsup_N\frac{1}{N} \sum_{n\le N}\mathbb{1}<em>E(T<sup>nx)=1$$ and $$\liminf_N\frac{1}{N} \sum</em>{n\le N}\mathbb{1}<em>E(T<sup>nx)=0$$ almost everywhere. In this paper we improve this result by showing that if $(a_n)$ satisfies only $\frac{a</em>{n+1}}{a_n}&gt;1+\frac1{(\log\log n)<sup>{1-\eta}}$ for some positive $\eta$ then it is already strong sweeping out.