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Claim. For every sequence $z_1,z_2,\ldots$ on the unit circle, with $s(k,n)=\sum_{j\le n}z_j^k$ and $A_k=\limsup_{n\to\infty}\lvert s(k,n)\rvert$, there are infinitely many $k$ with $A_k>c_1\log k$, for an absolute constant $c_1>0$; in particular $\limsup_kA_k=\infty$. This is the Theorem of Section 1 of P. Erdős, Some remarks on number theory, Israel J. Math. 3 (1965), 6-12 ([Er65c] on the problem page), pp. 6-7. With $z_j=e(x_j)$ it answers the first question of Problem 987 yes for every sequence in $(0,1)$: the paper's $A_k$ is the problem's $A_k$, a limit superior over $n$, which the paper distinguishes from $B_k$, the supremum over $n$ of the same sums. Erdős writes that in his 1964 paper ([Er64b] on the problem page) he had observed $\limsup_kB_k=\infty$ but stated that he could not prove the same for $A_k$, and that he had "overlooked the fact that it is very easy to show" the theorem.

Covers. The first question: $\limsup_{k\to\infty}A_k=\infty$ for every sequence, with the rate $\log k$. The rate was raised to $k^{1/2}$ by Clunie's theorem, which the paper's footnote added in proof already reports; 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.

Proof (pp. 6-7, read in full). Dirichlet's theorem on simultaneous approximation gives, for any $n$ complex numbers $y_1,\dots,y_n$ of modulus one, an integer $1\le k\le10^n$ with $\operatorname{Re}(y_i^k)>1/2$ for every $i$. Applied to the blocks $z_{rn+1},\dots,z_{(r+1)n}$ for $r=0,1,2,\dots$, this yields one $k\le10^n$ that serves infinitely many blocks, each of whose sums then exceeds $n/2$ in modulus, so $A_k\ge n/4$; since $k\le10^n$, this is $A_k>c_1\log k$, and letting $n$ grow gives infinitely many such $k$. The paper adds that $A_k\ge ck$ for infinitely many $k$ may hold, with $c\le1$ by a remark of Clunie that a footnote records, asks for the least $f(n,c)$ such that any $n$ complex numbers with $\lvert z_i\rvert\ge1$ have some $1\le k\le f(n,c)$ with $\lvert\sum_{i\le n}z_i^k\rvert\ge c$, notes $f(n,1)=n$ from Turán's results, and records in a footnote added in proof that Clunie proved $f(n,c)<g(c)\,n\log n$ and $A_k>ck^{1/2}$.

Source and dating. The page's basis is the scan in the Rényi Institute's Erdős archive, linked above. It was received on 10 February 1965 and appeared in volume 3, issue 1, which the publisher's record dates March 1965; the day in the page name is a placeholder. The site's commentary credits the proof to Erdős under its key [Er65b], which the site's reference record resolves to his 1965 Wiley lectures, which do not contain the passage; the Israel J. Math. note is the paper that holds it, as the problem page's reference [Er65c] records.

Formalization. The contributor's fork of formal-conjectures linked above states the theorem as erdos_987.variants.log_lower_bound, for sequences in $(0,1)$: there is $c>0$ with $c\log k\le A_k$ for infinitely many $k$. Its proof outline follows Erdős's argument, with simultaneous Dirichlet approximation to denominator $7^n$, the same block device and a pigeonhole on the blocks; the repository's main branch points the declaration's formal_proof attribute at that file. Not built or audited here, so it adds no evidence kind.

Acceptance. Refereed: Israel J. Math. 3 (1965), 6-12, a journal paper. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and credits Erdős in the problem's commentary with the proof that $A_k\gg\log k$ for infinitely many $k$ (page last edited 2026-04-09, as of 2026-10-07); Tao's thread comment of 2025-08-31 reports the same solution. The curator is independent of the author.