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 $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, 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.
Original language | English |
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Number of pages | 26 |
Publication status | Published - 2018 |