Erdős Problem #987

The problem

Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let

$$A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,$$

where $e(x)=e^{2\pi ix}$.

Is it true that

$$\limsup_{k\to \infty} A_k=\infty?$$

Is it possible for $A_k=o(k)$?

Statement quoted from erdosproblems.com/987, maintained by Thomas Bloom. Original source: [Er64b][Er65b][Ha74]. Current status there: proved (Lean).

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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