Erdős Problem #996
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #996 as open (status updated 7 September 2025).
- Formal Conjectures marks the Lean statement
erdos_996as open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #996 on erdosproblems.com | open 2025-09-07 | erdosproblems.com |
| Formal Conjectures: status of `erdos_996` | research open | Formal Conjectures |
| Formal Conjectures: research statements in the file | 1 open, 1 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-996,
title = {Erdős Problem #996},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-996}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-996)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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- Get a task — one click
- Your chatbot works on it — 5–10 minutes, in its own tab
- Paste its answer here — check it and publish it under your name
What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«996».
theorem erdos_996 :
∃ (C : ℝ), 0 < C ∧ ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
IsLacunary n →
(fun k => (eLpNorm (⇑f - fourierPartial f k) 2 (haarAddCircle (T := 1))).toReal) =O[atTop]
(fun k => 1 / (log (log (log k))) ^ C)
→
∀ᵐ x, Tendsto (fun N => (∑ k ∈ .range N, f (n k • x)) / N) atTop
(𝓝 (∫ t, f t ∂haarAddCircle))What counts as progress
- A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
- Partial results: special cases, weaker bounds, reductions — as verified claims.
- Computations and numerical evidence with published code (reproducible).
- Literature: the problem may have been solved or partly solved already — check erdosproblems.com/996. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
- erdosproblems.com/996
- [Er49d] Erdös, P. "On the strong law of large numbers." Transactions of the American Mathematical Society 67.1 (1949): 51-56.
- [Ma66] Matsuyama, Noboru. "On the strong law of large numbers." Tohoku Mathematical Journal, Second Series 18.3 (1966): 259-269.
Source and licence
Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.