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Submission note. Posted to the site's forum by Przemek Chojecki on 20 April 2026:

With GPT-5.4 Pro I've got a nice summary of literature related to the problem as well as partial results.

The claim. For $A\subseteq\{2,3,\ldots\}$ with $|A\cap[1,x]|=o(\sqrt x)$ let $F_A=\{b_1<b_2<\cdots\}$ be the integers divisible by no member of $A$ (so $b_1=1$), and for $h\ge1$ let $q_A(h)$ be the density of the $n$ for which the window $[n,n+h-1]$ contains no member of $F_A$. Theorem 1 of the note Erdős Problem #489: An extended-limit theorem, a finiteness criterion, and broad positive results: each $q_A(h)$ exists, and

$$\lim_{x\to\infty}\frac1x\sum_{b_i<x}(b_{i+1}-b_i)^2=1+2\sum_{h\ge1}q_A(h)$$

exists in $[0,+\infty]$, so the limit of Problem 489 exists once the value $+\infty$ is allowed, and it is finite exactly when the $q_A(h)$ are summable. Corollary 2: if $A$ is pairwise coprime and the multiplicative semigroup it generates has counting function $x^{\alpha+o(1)}$ for some $\alpha\in(0,1)$, then $q_A(h)\ll h^{-(2-\alpha)}$, so the limit is finite; this covers the squarefree numbers, for which Erdős proved the existence of the limit (the accepted partial claim [[problems/integer_sequences/E0489/claims/1951_05_04_erdos|Erdős's squarefree case]]), and the other classical sets of $k$-free numbers. The argument: the sparseness gives $\sum_{a\in A}1/a<\infty$, so $F_A$ has positive density and every fixed local pattern of $F_A$ has a frequency, by comparison with the periodic sieve by the members of $A$ up to a threshold; an exact identity writes the sum of squared gaps below $x$ as $x-1+r(x)^2+2\sum_h E_h(x)$ with $E_h(x)$ the number of empty windows of length $h$ starting below $x$ and $r(x)$ the overshoot to the next member of $F_A$; the overshoot is $o(\sqrt x)$, because a gap of length $G$ forces about $G$ distinct members of $A$ in $(G,u+G]$; and a dyadic truncation turns the identity into the formula. Corollary 2 applies a short-interval bound of Gorodetsky, Mangerel and Rodgers to the empty-window counts. The note closes with a literature survey and states that the unstructured case, finiteness under the sparseness hypothesis alone, remains open.

Covers. The existence of the limit in $[0,+\infty]$, with the formula above, for every $A\subseteq\{2,3,\ldots\}$ with $|A\cap[1,x]|=o(\sqrt x)$; and the finiteness asked for by the problem for every such $A$ that is pairwise coprime and generates a semigroup of index $\alpha\in(0,1)$. Not covered: finiteness in general, which the note identifies as the only remaining issue. The full claim on Snyder's page asserts finiteness for every such $A$.

Standing. Claimed. The note is linked from the author's comment of 20 April 2026 in the site's discussion thread, which says it was produced with GPT-5.4 Pro and describes it as a summary of the related literature with partial results; it is not on the proof-claim tab. A reply the same day reports that a check run with ChatGPT claimed several issues, confirms that the one concerning the bibliography is genuine, and recommends a revision; the author thanked the commenter. The site's label is OPEN, and no referee or named reviewer is on record. The argument was not checked here.

Depends on. Nothing in this wiki: the claim rests on its own note.