Problem 995

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$ . Estimate the growth of, for almost all $\alpha$ ,

\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).

For example, is it true that, for almost all $\alpha$ ,

\sum_{1\leq k\leq N}f(\{ \alpha n_k\})=o(N\sqrt{\log\log N})?

Source: erdosproblems.com/995

Analysis
Discrepancy

Wiki page

Status

Open

No claim settles this problem.

OPEN on erdosproblems.com; the site's commentary records Erdős's bounds and does not name a later result. One pending partial claim answers the example question in the negative: Boon Suan Ho's preprint of 20 April 2026, announced in the problem's thread on 21 April 2026, Ho 2026. No claim had been filed on the site's proof-claims tab.

Problem 995

Status

Proof claims

Discussion