# 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
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Tactic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Log
import Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

namespace JigCampaign.Dyadic

open MeasureTheory Set
open scoped ENNReal

def hit (m : ℕ) (c : ℝ) : Set ℝ :=
  Ico 0 1 ∩ {x | Int.fract ((m : ℝ) * x) < c}

theorem hit_eq_intervals {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1) :
    hit m c = ⋃ j ∈ Finset.range m, Ico ((j : ℝ) / m) ((j + c) / m) := by
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  ext x
  constructor
  · rintro ⟨hx, hh⟩
    change Int.fract ((m : ℝ) * x) < c at hh
    have hy : 0 ≤ (m : ℝ) * x := mul_nonneg hmR.le hx.1
    let j : ℕ := ⌊(m : ℝ) * x⌋₊
    have hj : j < m := (Nat.floor_lt hy).mpr (by nlinarith [hx.2])
    have hlow : (j : ℝ) ≤ (m : ℝ) * x := Nat.floor_le hy
    have hfract : Int.fract ((m : ℝ) * x) = (m : ℝ) * x - j := by
      rw [Int.fract, natCast_floor_eq_intCast_floor hy]
    refine mem_iUnion.mpr ⟨j, mem_iUnion.mpr ⟨Finset.mem_range.mpr hj, ?_⟩⟩
    constructor
    · apply (div_le_iff₀ hmR).mpr
      nlinarith
    · apply (lt_div_iff₀ hmR).mpr
      rw [hfract] at hh
      nlinarith
  · rintro hx
    obtain ⟨j, hx⟩ := mem_iUnion.mp hx
    obtain ⟨hj, hx⟩ := mem_iUnion.mp hx
    have hjm : j < m := Finset.mem_range.mp hj
    have hjR : (j : ℝ) + 1 ≤ m := by exact_mod_cast hjm
    have hlow : (j : ℝ) ≤ (m : ℝ) * x := by
      have := (div_le_iff₀ hmR).mp hx.1
      nlinarith
    have hupp : (m : ℝ) * x < j + c := by
      have := (lt_div_iff₀ hmR).mp hx.2
      nlinarith
    have hfloor : ⌊(m : ℝ) * x⌋ = (j : ℤ) :=
      Int.floor_eq_iff.mpr ⟨by exact_mod_cast hlow, by
        push_cast
        linarith⟩
    refine ⟨⟨?_, ?_⟩, ?_⟩
    · have : 0 ≤ (j : ℝ) := Nat.cast_nonneg j
      nlinarith
    · nlinarith
    · change Int.fract ((m : ℝ) * x) < c
      rw [Int.fract, hfloor]
      push_cast
      linarith

theorem grid_intervals_disjoint {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1) :
    Pairwise (fun i j : ℕ => Disjoint (Ico ((i : ℝ) / m) ((i + c) / m))
      (Ico ((j : ℝ) / m) ((j + c) / m))) := by
  intro i j hij
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  apply Set.disjoint_left.mpr
  intro x hi hj
  have hli := (div_le_iff₀ hmR).mp hi.1
  have hui := (lt_div_iff₀ hmR).mp hi.2
  have hlj := (div_le_iff₀ hmR).mp hj.1
  have huj := (lt_div_iff₀ hmR).mp hj.2
  rcases lt_or_gt_of_ne hij with h | h
  · have : (i : ℝ) + 1 ≤ j := by exact_mod_cast h
    linarith
  · have : (j : ℝ) + 1 ≤ i := by exact_mod_cast h
    linarith

theorem volume_hit {m : ℕ} (hm : 0 < m) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c ≤ 1) :
    volume (hit m c) = ENNReal.ofReal c := by
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  rw [hit_eq_intervals hm hc1]
  rw [measure_biUnion_finset (fun i _ j _ hij => grid_intervals_disjoint hm hc1 hij)
    (fun _ _ => measurableSet_Ico)]
  simp_rw [Real.volume_Ico, show ∀ j : ℕ, ((j : ℝ) + c) / m - j / m = c / m by
    intro j; ring]
  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
  rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg m)]
  congr 1
  field_simp

theorem volume_grid_union {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1)
    (S : Finset ℕ) :
    volume (⋃ j ∈ S, Ico ((j : ℝ) / m) ((j + c) / m)) =
      (S.card : ℝ≥0∞) * ENNReal.ofReal (c / m) := by
  rw [measure_biUnion_finset (fun i _ j _ hij => grid_intervals_disjoint hm hc hij)
-- 1322 more lines, see https://jig.so/p/
```

## 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
- 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
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Tactic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Log
import Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

namespace JigCampaign.Dyadic

open MeasureTheory Set
open scoped ENNReal

def hit (m : ℕ) (c : ℝ) : Set ℝ :=
  Ico 0 1 ∩ {x | Int.fract ((m : ℝ) * x) < c}

theorem hit_eq_intervals {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1) :
    hit m c = ⋃ j ∈ Finset.range m, Ico ((j : ℝ) / m) ((j + c) / m) := by
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  ext x
  constructor
  · rintro ⟨hx, hh⟩
    change Int.fract ((m : ℝ) * x) < c at hh
    have hy : 0 ≤ (m : ℝ) * x := mul_nonneg hmR.le hx.1
    let j : ℕ := ⌊(m : ℝ) * x⌋₊
    have hj : j < m := (Nat.floor_lt hy).mpr (by nlinarith [hx.2])
    have hlow : (j : ℝ) ≤ (m : ℝ) * x := Nat.floor_le hy
    have hfract : Int.fract ((m : ℝ) * x) = (m : ℝ) * x - j := by
      rw [Int.fract, natCast_floor_eq_intCast_floor hy]
    refine mem_iUnion.mpr ⟨j, mem_iUnion.mpr ⟨Finset.mem_range.mpr hj, ?_⟩⟩
    constructor
    · apply (div_le_iff₀ hmR).mpr
      nlinarith
    · apply (lt_div_iff₀ hmR).mpr
      rw [hfract] at hh
      nlinarith
  · rintro hx
    obtain ⟨j, hx⟩ := mem_iUnion.mp hx
    obtain ⟨hj, hx⟩ := mem_iUnion.mp hx
    have hjm : j < m := Finset.mem_range.mp hj
    have hjR : (j : ℝ) + 1 ≤ m := by exact_mod_cast hjm
    have hlow : (j : ℝ) ≤ (m : ℝ) * x := by
      have := (div_le_iff₀ hmR).mp hx.1
      nlinarith
    have hupp : (m : ℝ) * x < j + c := by
      have := (lt_div_iff₀ hmR).mp hx.2
      nlinarith
    have hfloor : ⌊(m : ℝ) * x⌋ = (j : ℤ) :=
      Int.floor_eq_iff.mpr ⟨by exact_mod_cast hlow, by
        push_cast
        linarith⟩
    refine ⟨⟨?_, ?_⟩, ?_⟩
    · have : 0 ≤ (j : ℝ) := Nat.cast_nonneg j
      nlinarith
    · nlinarith
    · change Int.fract ((m : ℝ) * x) < c
      rw [Int.fract, hfloor]
      push_cast
      linarith

theorem grid_intervals_disjoint {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1) :
    Pairwise (fun i j : ℕ => Disjoint (Ico ((i : ℝ) / m) ((i + c) / m))
      (Ico ((j : ℝ) / m) ((j + c) / m))) := by
  intro i j hij
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  apply Set.disjoint_left.mpr
  intro x hi hj
  have hli := (div_le_iff₀ hmR).mp hi.1
  have hui := (lt_div_iff₀ hmR).mp hi.2
  have hlj := (div_le_iff₀ hmR).mp hj.1
  have huj := (lt_div_iff₀ hmR).mp hj.2
  rcases lt_or_gt_of_ne hij with h | h
  · have : (i : ℝ) + 1 ≤ j := by exact_mod_cast h
    linarith
  · have : (j : ℝ) + 1 ≤ i := by exact_mod_cast h
    linarith

theorem volume_hit {m : ℕ} (hm : 0 < m) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c ≤ 1) :
    volume (hit m c) = ENNReal.ofReal c := by
  have hmR : 0 < (m : ℝ) := by exact_mod_cast hm
  rw [hit_eq_intervals hm hc1]
  rw [measure_biUnion_finset (fun i _ j _ hij => grid_intervals_disjoint hm hc1 hij)
    (fun _ _ => measurableSet_Ico)]
  simp_rw [Real.volume_Ico, show ∀ j : ℕ, ((j : ℝ) + c) / m - j / m = c / m by
    intro j; ring]
  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
  rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg m)]
  congr 1
  field_simp

theorem volume_grid_union {m : ℕ} (hm : 0 < m) {c : ℝ} (hc : c ≤ 1)
    (S : Finset ℕ) :
    volume (⋃ j ∈ S, Ico ((j : ℝ) / m) ((j + c) / m)) =
      (S.card : ℝ≥0∞) * ENNReal.ofReal (c / m) := by
  rw [measure_biUnion_finset (fun i _ j _ hij => grid_intervals_disjoint hm hc hij)
-- 1322 more lines, see https://jig.so/p/
```

- Canonical statement

```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
```

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