The discrepancy of $(n_{k}x)$ with respect to certain probability measures

Abstract

Let $(n_k{)}_{k=1}^{\infty }$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\hat{\mu }(t)|\le c|t{|}^{-\eta }$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_{k}x)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\Vert q\alpha \Vert \Vert q\beta -\gamma \Vert$.