DISPROVED
This has been solved in the negative.
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\[D(N)\ll N^{1/2}(\log\log N)^{O(1)}?\]
Erdős and Koksma
[ErKo49] and Cassels
[Ca50] independently proved that, for any sequence $x_i$ and almost all $\alpha$, the discrepancy satisfies\[D(N)\ll N^{1/2}(\log N)^{5/2+o(1)}.\]Baker
[Ba81] improved this to\[D(N)\ll N^{1/2}(\log N)^{3/2+o(1)}.\]Erdős and Gál (unpublished) proved $D(N) \ll N^{1/2}(\log\log N)^{O(1)}$ for almost all $\alpha$ if the sequence is lacunary - that is, $x_{i+1}/x_i > \lambda>1$ for all $i$.
This was disproved by Berkes and Philipp
[BePh94], who constructed a sequence of integers $x_1<x_2<\cdots$ such that, for almost all $\alpha\in[0,1]$,\[\limsup_{N\to \infty}\frac{D(N)}{(N\log N)^{1/2}}>0.\]
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This page was last edited 29 January 2026. View history
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Additional thanks to: Superhuman Reasoning team at Google DeepMind
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #992, https://www.erdosproblems.com/992, accessed 2026-04-11
