Problem 994

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claims/: The 1 claim page of Problem 994, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $E$?

Statement (precise). Let $E\subseteq (0,1)$ be a meaurable subset with Lebesgue measure $\lambda(E)$. Is it true that, for all $E$,

$$\lim_{n\to \infty}\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}=\lambda(E)$$

for almost all $\alpha$?

Notes. The site's wording places "for all $E$" after "for almost all $\alpha$", so it also admits a simultaneous reading, in which one set of $\alpha$ of full measure serves every measurable $E$ at once. That reading fails for every $\alpha$: the orbit $\{\{k\alpha\}:k\ge1\}$ is countable, so its complement in $(0,1)$ is measurable, has measure $1$ and is never visited, and its visit frequency is $0$; the argument is elementary and is set out on the claim page. The change swaps the two phrases "for almost all $\alpha$" and "for all $E$", so that the null set of exceptional $\alpha$ may depend on $E$; nothing else changes. The evidence is the poser's own text as the site credits it: the site's commentary calls the problem a conjecture of Khintchine [Kh23], and Khintchine's question (§ 5, "Ein neues Problem", pp. 303–304) fixes the set $E$ first and asks whether his relations (6) and (7) hold "für alle $x$ mit Ausnahme höchstens einer Menge vom Maße Null". The ambiguity is already in Erdős's statement in [Er64b] (Part II, p. 57), which the site's wording follows: "Then for almost all $\alpha$ and every $E$". Erdős's own words there point to the same reading, since he credits the conjecture to Khintchine with the locator "see p. 303–304" and calls it "very deep", which the simultaneous reading is not. The choice does not change the answer: Marstrand's refutation [Ma70] of the precise Statement refutes the simultaneous reading as well. Results about the simultaneous reading alone, credited here and not counted: the variant erdos_994.variants.simultaneous of the formal-conjectures statement file (added on 2026-09-22, pinned under Formalization) and the theorem not_erdos_994 of the file Erdos994.lean in Boris Alexeev's lean-proofs collection (in the repository since 2026-08-17; formal authors Codex and GPT-5.6 Sol, as the file names them), both proving the orbit argument in Lean.

Status. DISPROVED (LEAN), the label the community database that the site displays lists as of its last update on 2026-09-16 (the site's label on 2026-09-04 was DISPROVED): Khintchine's question of 1923 [Kh23], which Erdős's 1964 problem paper [Er64b] calls a conjecture, was refuted by Marstrand [Ma70], so the precise Statement is false, and the site's wording is false in either order of its two quantifiers, as the accepted claim Marstrand 1970 explains. The Lean behind the qualifier is a third-party formalization of Marstrand's disproof in the fixed-set order, not built or audited in this corpus.

Source. erdosproblems.com/994, accessed 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #994, https://www.erdosproblems.com/994.

References.

• [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. 16 (1964), 52-65; Part II, p. 57. Library home: erdos_1964_problems_results_diophantine_approximations.
• [Kh23] Khintchine, A., Ein Satz über Kettenbrüche, mit arithmetischen Anwendungen. Math. Z. (1923), 289-306; § 5, pp. 303-304. Library home: khintchine_1923_ein_satz_uber_kettenbruche_mit.
• [Ma70] Marstrand, J. M., On Khinchin's conjecture about strong uniform distribution. Proc. London Math. Soc. (3) (1970), 540-556.

Formalization. Statement in formal-conjectures, added on 2026-09-22 and pinned to that commit: its main theorem erdos_994, in the fixed-set order of the precise Statement, is tagged research solved with the answer False and left without proof, and its variant erdos_994.variants.simultaneous proves in Lean that the simultaneous reading of the site's wording is false, by removing the countable orbit of $\alpha$ from $(0,1)$. The Lean behind the site's qualifier is Collin Yuanjie Ren's formalization of Marstrand's disproof in the fixed-set order (2026-09-16), which the community database cites; a file in Boris Alexeev's lean-proofs collection proves the simultaneous reading false. All three are linked, pinned, on the claim page; none is built or audited in this corpus.

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_1964_problems_results_diophantine_approximations
• khintchine_1923_ein_satz_uber_kettenbruche_mit
• khintchine_1923_ein_satz_uber_kettenbruche_mit / problem_p303
• khintchine_1923_ein_satz_uber_kettenbruche_mit / theorem_p289
• khintchine_1923_ein_satz_uber_kettenbruche_mit / theorem_p298
• khintchine_1923_ein_satz_uber_kettenbruche_mit / theorem_p305