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?

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.

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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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_996 : answer(sorry) ↔
    ∃ (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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.