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Claim. On 4 May 2026 the forum user aditya posted in the thread of Problem 332 an observation attributed to GPT 5.5 pro, as the post names the system: the positive-density condition can be weakened to positive upper Banach density. The post links a three-page note, A Banach-density condition for bounded gaps in an infinite difference set, whose author line is a placeholder, so it names no author. Its theorem: if $d^*(A)=\limsup_{L\to\infty}\sup_M|A\cap[M,M+L)|/L>0$, then $D(A)$ has bounded gaps; equivalently the two-sided set $\Delta(A)$ of $t$ with $A\cap(A-t)$ infinite is syndetic in $\mathbb Z$. The proof: translates $A-f$, $f\in F$, whose pairwise differences avoid $\Delta(A)$ meet pairwise in finite sets, so counting on intervals where $A$ has density near $d^*(A)$ gives $|F|\le1/d^*(A)$, and a maximal such $F$ satisfies $\mathbb Z=F+\Delta(A)$. The same conclusion follows from Theorem 4.1 of Belgikar, Bergelson, Black and Kruzel (arXiv:2412.01185v2, 2025), which allows any Følner sequence, applied along intervals on which $A$ has density tending to $d^*(A)$ (an observation made here).

Covers. Every $A\subseteq\mathbb N$ of positive upper Banach density, which includes every set of positive upper density (Stewart and Tijdeman's theorem). Not covered: sets of upper Banach density zero.

Standing. Claimed: an unrefereed note that names no author and has no recorded review; the site labels the problem OPEN and its proof-claims tab is empty.

Depends on. No page of this wiki.