  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 $C>0$,
proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight
improvement on their previous result on products of the form $q\|q\alpha\|
\|q\beta-\gamma\| $.

Number Theory