Problem 996

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$ . Let $f_n$ be the $n$ th partial sum of the Fourier series of $f(x)$ . Is there an absolute constant $C>0$ such that, if

\| f-f_n\|_2 \ll \frac{1}{(\log\log\log n)^{C}}

then

\lim_{N\to\infty}\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})=\int_0^1 f(x)\mathrm{d}x

for almost every $\alpha$ ?

Source: erdosproblems.com/996

Analysis

Wiki page

Status

ClaimedDisproved

A full solution has been claimed but not yet accepted. The statement is false.

Open. The site's proof-claims tab carries one full proof claim, filed 26 September 2026 by Ethan Yang, using GPT-6 Astra and GPT-5.6 Sol as the tab names them, and a thread comment of 27 April 2026 points to Boon Suan Ho's earlier preprint; the site's label is unchanged (OPEN). The pending claims, both negative answers, are Ho's page and Yang's page.

Problem 996

Status

Proof claims

Discussion