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Claim. For every sequence $x_1,x_2,\ldots\in(0,1)$, with $A_k=\limsup_n\lvert\sum_{j\le n}e(kx_j)\rvert$, there are infinitely many $k$ with $A_k\gg k^{1/2}$; in particular $\limsup_k A_k=\infty$. The same paper shows that some sequences have $A_k\le k$ for every $k$, a bound that is not $o(k)$ and so leaves the second question of Problem 987 open.

Covers. The first question: $\limsup_{k\to\infty}A_k=\infty$ for every sequence, with the rate $k^{1/2}$ in place of Erdős's $\log k$. The second question, whether $A_k=o(k)$ is possible, is answered by the 2026 construction.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Earlier and later bounds. The site records that Erdős remarked in 1964 that the variant with $\sup_n$ in place of $\limsup_n$ is easily seen to be unbounded in $k$, and that in 1965 he gave a short proof of $A_k\gg\log k$ for infinitely many $k$, which already answers the first question (Erdős's claim page); Clunie's bound is the first of power type. For sequences that take only finitely many distinct values, Liu's 1969 bound $A_k>k^{1-\delta}/5$ for infinitely many $k$ is on Liu's claim page. The 2025 forum proof and Lean formalization of the first question are on Tao's claim page.

Dating. The page is dated by the publication year. The journal record (J. London Math. Soc. 42 (1967), 133–136) gives no day, and the day in the page name is a placeholder.

Formalization. The contributor's fork of formal-conjectures linked above states Clunie's bound as erdos_987.variants.sqrt_lower_bound and his linear upper bound as erdos_987.variants.linear_upper_bound, with proofs its comments attribute to the lemmas of the 1967 paper; the repository's main branch points its formal_proof attributes at that file. Not built or audited here, so it adds no evidence kind.

Acceptance. Refereed: J. London Math. Soc. 42 (1967), 133–136. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and credits Clunie in the problem's commentary with $A_k\gg k^{1/2}$ for infinitely many $k$ and with sequences having $A_k\le k$ for all $k$ (page last edited 2026-04-09). The curator is independent of the author. The statements above follow the site's record of the paper.