Erdős Problem #995

The problem

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 $\alpha$,

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

Statement quoted from erdosproblems.com/995, maintained by Thomas Bloom. Original source: [Er64b]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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