Erdős (Erdos) Problem #995: status in 2026

Erdős Problem #995

Erdős Problem 995: For every lacunary sequence (n_k) of integers and every f ∈ L^2([0,1]) with ∫_0^1 f = 0, is it true that for almost all α, Σ_k < N f(α n_k) = o (N √(loglog N))?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-07).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #995 as open (status updated 7 September 2025).
Formal Conjectures marks the Lean statement erdos_995 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #995 on erdosproblems.com	open 2025-09-07	erdosproblems.com
Formal Conjectures: status of `erdos_995`	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.

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Cite

@misc{cairn-erdos-995,
  title        = {Erdős Problem #995},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-995}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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.

The problem

The question

Erdős Problem 995:

For every lacunary sequence $(n_{k})$ of integers and every $f \in L^{2} ([0, 1])$ with $\int_{0}^{1} f = 0$ , is it true that for almost all $α$ , $k < N \sum f ({α n_{k}}) = o (N lo g lo g N) ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«995».

theorem erdos_995 :
    ∀ (n : ℕ → ℕ), IsLacunary n → ∀ (f : ℝ → ℝ),
      MemLp f 2 (volume.restrict (Icc (0 : ℝ) 1)) →
      ∫ x in (0 : ℝ)..1, f x = 0 →
      ∀ᵐ α ∂(volume.restrict (Icc (0 : ℝ) 1)),
        partialSum n f α =o[atTop] fun N => (N : ℝ) * Real.sqrt (Real.log (Real.log N))

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/995. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/995
[Er49] Erdős, P., On the strong law of large numbers, Trans. Amer. Math. Soc. (1949), 329-334.

Let $n_{1} < n_{2} < \dots$ be a lacunary sequence of integers and $f \in L^{2} ([0, 1])$ with $\int_{0}^{1} f = 0$ . Estimate the growth of $S_{N} (α) = k \leq N \sum f ({α n_{k}})$ for almost all $α$ , where ${\cdot}$ denotes the fractional part. In particular, is it true that $S_{N} (α) = o (N lo g lo g N)$ for almost all $α$ ?

Erdős [Er49] constructed a lacunary sequence and a mean-zero $f \in L^{2} ([0, 1])$ for which, for every $ε > 0$ , $lim sup_{N} S_{N} (α) / (N (lo g lo g N)^{1/2 - ε}) = \infty$ for almost all $α$ , and believed this lower bound to be close to the truth.

The mean-zero hypothesis $\int_{0}^{1} f = 0$ is the natural normalisation: otherwise the sum has a linear main term coming from the average of $f$ .

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.