Erdős Problem #996

Erdős problem #996 · erdosproblems.com/996

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$ with $n$th Fourier partial sum $f_n$. Is there an absolute constant $C>0$ such that if $\| f-f_n\|_2 \ll (\log\log\log n)^{-C}$ then $\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f$ for almost every $\alpha$? A preprint answers this negatively via a dyadic spike-block counterexample.

Result

Disproved(see note)

Status

Candidate (review pending)

AI contribution

AI-assisted

Method

Construction

Field

Analysis, Fourier Series

Posed by

Paul Erdős

Year posed

1964

Years open

62y

Solved

2026-04-21

Model

GPT-5.4 Pro

Vendor

OpenAI

Collaborators

Boon Suan Ho

Verification

Unreviewed

Publication

Preprint

Significance

10 / 100*

Disclosed cost

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Wikipedia

No dedicated article