Erdős Problem #994
DISPROVED LEANdatabase updated 2026-09-16
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The proportion \(\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}\) of the fractional parts \(\{k\alpha\}\), \(1\leq k\leq n\), which lie in `E`.
Let \(E\subseteq (0,1)\) be a meaurable subset with Lebesgue measure \(\lambda(E)\). Is it true that, for almost all \(\alpha\), \[\lim_{n\to \infty}\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}=\lambda(E)\] for all \(E\)? This is a conjecture of Khintchine [Kh23] (with the exceptional null set of \(\alpha\) allowed to depend on \(E\)). It is false, and was disproved by Marstrand [Ma70].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/994.Prizenone
Topicsanalysis, discrepancy
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/994
Attempts
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