# Jig #265: Prior art

> Can lacunary L² sums exceed N sqrt(log log N) almost surely?
>
> [arXiv:2604.18535v2](https://arxiv.org/abs/2604.18535v2)

- URL: https://jig.so/p/265
- Status: Prior art
- Erdős problem: 995 (https://www.erdosproblems.com/995)
- Prior art: https://arxiv.org/abs/2604.18535v2 — Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks, Theorem1.6 atp=2 (and its explicit consequence forN sqrt(loglogN)), Sections2–4 and7. The complete root is known; this submission formalizes a nonnegative integer-base variant and claims no novelty.
- Posed: 2026-08-25T07:33:30.934Z
- Last statement: 2026-08-25T07:33:30.937Z
- Last activity: 2026-09-07T01:50:01.181Z
- Statements: 1
- Contributors: @woshuajolk, @hd1932

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: Lacunarity is the Hadamard gap condition n_(k+1)≥q n_k for some q>1 and a positive integer sequence. L² is with respect to Lebesgue measure on [0,1], arguments are reduced to fractional parts, and almost every alpha is Lebesgue almost everywhere. The canonical polarity follows Ho’s 2026 counterexample, not the stale affirmative question.
- By: @hd1932, @woshuajolk

- NonnegativeSpikes.lean: Submissions.Erdos995LacunaryL2Growth.NonnegativeSpikes.proof — axioms clean

```lean
/-
Complete Jig265 counterexample, after Boon Suan Ho arXiv2604.18535v2.
Fresh Lean formalization of a nonnegative integer-base variant of the spike
construction. Exact cylinder measures, second-moment estimates, square-
integrability and the final canonical negation are all proved in this file.
No mathematical novelty is claimed.
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.MeasureTheory.Function.L2Space
-- clipped, full source: https://jig.so/api/artifacts/04a3af2a-4dd6-46c0-bee1-033abe639100/source
```

### Verification

Read from the verifier's report on the closing proof.

- Verdict report: [artifact 04a3af2a](https://jig.so/api/artifacts/04a3af2a-4dd6-46c0-bee1-033abe639100)
- CI run: [GitHub Actions log](https://github.com/WoshuaJolk/jig-verifier/actions/runs/34074279003)
- Checks passed: build, axioms, manifest, no new axioms, static policy, anti restatement
- Axioms: Classical.choice, Quot.sound, propext
- Toolchain: Lean v4.33.0, Mathlib db584cd (Jig's pins)

## Statements (1)

### 1. The proposed universal almost-sure o(N sqrt(log log N)) bound for lacunary dilates of arbitrary L² functions…

- Permalink: https://jig.so/p/265?s=1
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/04a3af2a-4dd6-46c0-bee1-033abe639100 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34074279003, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T07:33:30.000Z by @woshuajolk, @hd1932 / GPT 5.6 Sol / Cursor
- Version: 3
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The proposed universal almost-sure o(N sqrt(log log N)) bound for lacunary dilates of arbitrary L² functions is false.**

**Scope.**

Lacunarity is the Hadamard gap condition n_(k+1)≥q n_k for some q>1 and a positive integer sequence. L² is with respect to Lebesgue measure on [0,1], arguments are reduced to fractional parts, and almost every alpha is Lebesgue almost everywhere. The canonical polarity follows Ho’s 2026 counterexample, not the stale affirmative question.

**Artifacts.**

- NonnegativeSpikes.lean: Submissions.Erdos995LacunaryL2Growth.NonnegativeSpikes.proof

```lean
/-
Complete Jig265 counterexample, after Boon Suan Ho arXiv2604.18535v2.
Fresh Lean formalization of a nonnegative integer-base variant of the spike
construction. Exact cylinder measures, second-moment estimates, square-
integrability and the final canonical negation are all proved in this file.
No mathematical novelty is claimed.
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.MeasureTheory.Function.L2Space
-- clipped, full source: https://jig.so/api/artifacts/04a3af2a-4dd6-46c0-bee1-033abe639100/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Order.Filter.AtTopBot.Basic

open Filter MeasureTheory Set
open scoped BigOperators ENNReal MeasureTheory

namespace Statements.Erdos995LacunaryL2Growth

def IsLacunary (n : ℕ → ℕ) : Prop :=
  1 ≤ n 0 ∧ ∃ q : ℝ, 1 < q ∧ ∀ k : ℕ,
    q * n k ≤ n (k + 1)

noncomputable def normalizedSum (n : ℕ → ℕ) (f : ℝ → ℝ)
    (α : ℝ) (N : ℕ) : ℝ :=
  (∑ k ∈ Finset.range N, f (Int.fract (α * n k))) /
    ((N : ℝ) * Real.sqrt (Real.log (Real.log N)))

/-- Erdős 995 has a negative answer: the proposed universal almost-sure
`o(N * sqrt(log log N))` bound fails for some `L²` function and
lacunary sequence. -/
abbrev statement : Prop :=
  ¬ ∀ n : ℕ → ℕ, ∀ f : ℝ → ℝ,
    IsLacunary n →
    MemLp f 2 (volume.restrict (Icc (0 : ℝ) 1)) →
    ∀ᵐ α : ℝ ∂volume,
      Tendsto (normalizedSum n f α) atTop (nhds 0)

theorem target : statement := sorry

end Statements.Erdos995LacunaryL2Growth
```

## Contributing

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