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Claim. For the function $f$ of Problem 169, the supremum of $\sum_{n\in A}1/n$ over sets $A$ of positive integers with no $k$-term arithmetic progression,

$$f(k)\ge(1-o(1))\,k\log k\qquad(k\to\infty).$$

This is Gerver's theorem, which the site's commentary credits to him and which Walker's introduction records in the same form; it improves the linear bound of Berlekamp. The same paper proves that the finiteness of $f(k)$ for every $k$ is equivalent to Erdős's conjecture that a set with divergent reciprocal sum contains progressions of every length, Problem 3; that equivalence is a reduction and settles no instance of the estimate, so it stays in prose. Problem 3 is solved in this corpus on its claim page, so by the equivalence every $f(k)$ is finite; the explicit form of a bound is the subject of the release's claim page.

Covers. The lower bound $f(k)\ge(1-o(1))k\log k$. Not covered: any upper bound or estimate of $f(k)$, and the displayed limit question, since $\log W(k)$ is not known to grow slower than $k\log k$.

Depends on. No page of this wiki; the result is the paper's.

Acceptance. Refereed: J. L. Gerver, The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms, Proc. Amer. Math. Soc. 62 (1977), no. 2, 211–214. The site's curator credits the bound in the problem's commentary, but the site labels the problem OPEN, so that commentary is not acceptance of the problem and the page lists no reviewed evidence.

Dating. The page is dated by the issue month in the publisher's record, February 1977; the day is a placeholder.