Problem 987

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 $A_k=o(k)$ ?

Source: erdosproblems.com/987

Analysis
Discrepancy

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN), the site's label: the first question was answered by Erdős himself in 1965, with the bound $A_k>c\log k$ for infinitely many $k$ (Erdős's 1965 theorem), sharpened to $A_k\gg k^{1/2}$ by Clunie's 1967 theorem, the first bound of power type, and reproved, with a Lean formalization, on Tao's 2025 claim page (2025); the second is answered by the 2026 construction of Alexeev, Putterman, Sawhney, Sellke and Valiant, an unrefereed preprint accepted by the site's curator, Thomas Bloom. For sequences with finitely many distinct values, Liu's 1969 theorem gives $A_k>k^{1-\delta}/5$ for infinitely many $k$ ; it settles neither question as a whole. Each of the other four claims covers one of the two questions, so each is an accepted partial claim; the two questions are the problem's two parts, each part is settled by accepted partial claims, and the frontmatter standing derives from them as solved and proved, the two answers counted together as the site's label counts them. The site's Lean qualifier matches the community database's Lean status, dated 2026-08-23, which Boris Alexeev set in a batch of forty problems whose solutions his lean-proofs collection formalizes; that collection's file for this problem proves both questions (see Formalization).

APSSV26b: B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics, probability, and number theory II. arXiv:2604.06609 (2026).
Cl67: Clunie, J., On a problem of Erdős. J. London Math. Soc. (1967), 133-136.
Er64b: Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.
Er65b: Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
Er65c: Erdős, P., Some remarks on number theory. Israel J. Math. 3 (1965), no. 1, 6-12, DOI 10.1007/BF02760020; received 10 February 1965. Section 1, pp. 6-7; the scan in the Rényi Institute's Erdős archive is linked from Erdős's claim page. The site's commentary credits the $\log k$ bound to Erdős under its key [Er65b], which the site's reference record resolves to the lectures above, which do not contain the passage; this note is the paper that holds it, the 1965 Israel J. Math. note that [Liu69] cites, and the formal-conjectures file's reference list names it under the site's key.
Ha74: Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
Liu69: Liu, Ming-chit, On a problem of Erdős. Proc. Amer. Math. Soc. 21 (1969), 706-710, DOI 10.1090/S0002-9939-1969-0245795-8; MathSciNet review MR0245795. Received 12 August 1968; the theorem is on p. 706. The site's commentary names Liu under its key [Li69], which the site's reference record resolves to Lindström, B., An inequality for $B_2$ -sequences, J. Combinatorial Theory (1969), 211-212, the entry it shares with Problem 30 and a paper that does not concern this problem; the record above is the paper the commentary describes, which cites [Cl67], [Er64b] and Erdős's 1965 Israel J. Math. note.

Problem 987

Status

Proof claims

Discussion