The Discrepancy of (nkx)k=1 with Respect to Certain Probability Measures

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 $|\widehat{\mu }(t)|\leq c|t|^{-\eta }$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ 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 $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood conjecture.