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Erdős problem 996

Let n1<n2<⋯n_1<n_2<\cdots be a lacunary sequence of integers and f∈L2([0,1])f\in L^2([0,1]) with nnth Fourier partial sum fnf_n. Is there an absolute constant C>0C>0 such that if ∥f−fn∥2≪(log⁡log⁡log⁡n)−C\| f-f_n\|_2 \ll (\log\log\log n)^{-C} then 1N∑k≤Nf({αnk})→∫01f\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f for almost every α\alpha? A preprint answers this negatively via a dyadic spike-block counterexample.

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Retained declaration

FormalConjectures/ErdosProblems/996.lean

Formal Conjectures

FormalConjectures/ErdosProblems/996.leanErdos996.erdos_99611 linesExact file
sorry ↔  ∃ C,    0 < C ∧      ∀ (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (n : ℕ → ℕ),        IsLacunary n →          ((fun k =>                (MeasureTheory.eLpNorm (Erdos996.fourierPartial f k) 2 AddCircle.haarAddCircle).toReal) =O[Filter.atTop]              fun k => 1 / Real.log (Real.log (Real.log ↑k)) ^ C) →            ∀ᵐ (x : AddCircle 1),              Filter.Tendsto (fun N => (∑ k ∈ Finset.range N, ↑↑f (n k • x)) / ↑N) Filter.atTop                (nhds (∫ (t : AddCircle 1), ↑↑f t ∂AddCircle.haarAddCircle))
OpenStatement only, no proof

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Work these sources record against this Problem. Source-reported attribution, not reviewed here.

  • AI collaborating with humans

    Erdős AI contributions wiki · 21 Apr, 2026

    Machine
    GPT-5.4 Pro
    People
    Boon Suan Ho
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  • construction

    VibeMathed

    Machine
    GPT-5.4 Pro
    People
    Boon Suan Ho
    Reported outcome
    candidate
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