Erdős Problem #992

declared status 'disproved'. Formalized: no. 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$$D(N)\ll N^{1/2}(\log\log N)^{O(1)}?$$ Prize: no. Tags: discrepancy.

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