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Claim. The Theorem of M.-C. Liu, On a problem of Erdős, Proc. Amer. Math. Soc. 21 (1969), 706--710 (p. 706): if a sequence $z_1,z_2,\ldots$ on the unit circle takes only finitely many distinct values, then for every $\delta>0$, $\limsup_{k\to\infty}A_k/k^{1-\delta}>1/5$, where $A_k=\limsup_n\lvert\sum_{\nu\le n}z_\nu^k\rvert$ (Liu's $B_k$). With $z_j=e(x_j)$ this answers the first question of Problem 987 yes for every sequence in $(0,1)$ with finitely many distinct terms, with $A_k>k^{1-\delta}/5$ for infinitely many $k$. The site's commentary adds that Clunie's MathSciNet review of the paper (MR0245795) observes that $A_k$ is infinite for infinitely many $k$ under the same hypothesis. formal-conjectures states that observation as erdos_987.variants.finite_distinct_points, which the contributor's fork, linked on the other claim pages, proves.

Covers. The first question for sequences with finitely many distinct values. The question for every sequence is answered on Erdős's and Clunie's claim pages.

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

Dating. The paper appeared in volume 21, number 3, the June 1969 issue; the day in the page name is a placeholder.

Acceptance. Refereed: Proc. Amer. Math. Soc. 21 (1969), no. 3, 706--710. The site's commentary credits Liu under its key [Li69], which its reference record resolves to a different paper; the problem page's reference [Liu69] records the publisher's citation.