---

Claim. For a base $b>2$ and a proper subset $S\subset\{0,\ldots,b-1\}$, the Kempner set $\mathcal K(S,b)$ is the set of nonnegative integers all of whose base-$b$ digits lie in $S$. Walker's paper gives conditions under which a Kempner set has no $k$-term arithmetic progression and, by a search over such sets, finds that $\mathcal K(\{0,1,2,4,5,9,10,11,14,16,17,18,21,24,30,37,39,41,42,45,47\},55)+1$ has no four-term progression and reciprocal sum $4.43975$, and that an explicit Kempner set to base $77$ has no ten-term progression and reciprocal sum $14.056$. For the function $f$ of Problem 169, which the paper writes $M_k$,

$$f(4)\ge4.43975\qquad\text{and}\qquad f(10)\ge14.056,$$

the latter improving $13.5905$ from the set $(G_7+3)\cup\{1,2,3\}$, a translate of the greedy set $G_7$ with $1,2,3$ adjoined. The site's commentary credits $f(4)\ge4.43975$ to Walker. The paper's Theorem 2.1 also shows that Kempner sets suffice: for every $k\ge3$ and $\varepsilon>0$ there is a Kempner set with no $k$-term progression whose reciprocal sum exceeds $f(k)-\varepsilon$. That reduction settles no instance of the estimate, so it stays in prose. Kiichi's explicit set of 2026, built by gluing blocks onto Walker's base-$55$ set, claims the larger value $f(4)\ge4.4397534742$ on its own page.

Covers. The lower bounds $f(4)\ge4.43975$ and $f(10)\ge14.056$. Not covered: the values of $f(4)$ and $f(10)$, any other $k$, and the displayed limit question.

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

Standing. Claimed. The paper is an arXiv preprint (version 1 of 11 March 2022, version 2 of 4 September 2025) with no journal record, so the page lists no refereed evidence; 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.