Erdős Problem #992 — Let $x_1<x_2<\cdots$ be an infinite sequence of integers.

Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$, the discrepancy $D(N)=\max_{I\subseteq [0,1]} \lvert \#\{ n\leq N : \{ \alpha x_n\}\in I\} - \lvert I\rvert N\rvert$ satisfies $D(N) \ll N^{1/2}(\log N)^{o(1)}?$ Or even $D(N)\ll N^{1/2}(\log\log N)^{O(1)}?$