Problem 992

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claims/: The 1 claim page of Problem 992, one per claimant's result; the problem's standing derives from them.

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Statement. 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

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

Status. Disproved: Berkes and Philipp [BePh94] built an integer sequence whose discrepancy has $\limsup D(N)/(N\log N)^{1/2}>0$ for almost every $\alpha$, so both asked bounds fail; the accepted claim is Berkes and Philipp 1994.

Source. erdosproblems.com/992, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #992, https://www.erdosproblems.com/992.

References.

• [Ba81] Baker, R. C., Metric number theory and the large sieve. J. London Math. Soc. (2) (1981), 34-40.
• [BePh94] Berkes, István and Philipp, Walter, The size of trigonometric and Walsh series and uniform distribution ${\rm mod}\ 1$. J. London Math. Soc. (2) (1994), 454-464.
• [Ca50] Cassels, J. W. S., Some metrical theorems of Diophantine approximation. III. Proc. Cambridge Philos. Soc. (1950), 219-225.
• [ErKo49] Erdős, P. and Koksma, J. F., On the uniform distribution modulo $1$ of sequences $(f(n,\theta))$. Nederl. Akad. Wetensch., Proc. (1949), 851-854 = Indagationes Math. 11, 299-302.

Formalization. None built or audited here. A public Lean 4 development in Boris Alexeev's lean-proofs collection declares itself a formalization of Berkes and Philipp's disproof, a self-contained version of their mechanism, and is linked, pinned, on the claim page. The site records no formal-conjectures statement file, and the community database lists the problem as unformalized.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1949_uniform_distribution_modulo_1_lacunary_sequences
• erdos_1949_uniform_distribution_modulo_1_lacunary_sequences / theorem_1
• feng_2026_semi_autonomous_mathematics_discovery_gemini_case