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Erdős Problem #996

Analysis · posed by Paul Erdős, 1964 · candidate

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Statement

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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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Boon Suan Ho, using GPT-5.4 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    ai assisted
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    Per the paper's acknowledgements, GPT-5.4 Pro was used during development to explore proof strategies, test intermediate formulations and assist with exposition; all arguments were independently verified by the author, who takes full responsibility.

    Answered negatively in a preprint that also settles the p=2 case of problem #995; erdosproblems.com still lists the problem open

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