Erdős Problem #992

The problem

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

$$D(N)\ll N^{1/2}(\log\log N)^{O(1)}?$$

Statement quoted from erdosproblems.com/992, maintained by Thomas Bloom. Original source: [Er64b,p.56]. Current status there: disproved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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