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Claim. Theorem 1.4 of E. Ackelsberg and F. K. Richter, An inverse theorem for sumsets of sets of positive density in the integers, arXiv:2604.12864 (14 April 2026), cited as [AcRi26] on the problem page and digested on the library card ackelsberg_2026_inverse_theorem_sumsets_sets_positive_density, characterizes the pairs of Problem 335 in which one set meets every residue class. Let $A,B\subseteq\mathbb N$ with $d(A)>0$ and $d(A)+d(B)<1$, and let $B$ meet every residue class, that is, $B\cap(a\mathbb N+b)\ne\varnothing$ for all $a,b\in\mathbb N$. If $d(A+B)=d(A)+d(B)$ along some sequence of scales $N_s\to\infty$, the density of $A+B$ being taken along that sequence, then for some $h$ there is a decomposition $A=A_0-a_0$ and $B=(B_0\cup B_1)-b_0$ with $A_0,B_0\subseteq h\mathbb N$, $a_0,b_0\in\{0,\ldots,h-1\}$ and $B_1\subseteq\mathbb N\setminus h\mathbb N$, so that $A$ lies in one residue class modulo $h$, and one of two cases holds. In case (1), $B_1$ contains all of $\mathbb N\setminus h\mathbb N$ up to a set of density zero, and there are an irrational $\theta$ and closed intervals $I,J$ of the circle such that, with $\phi(n)=n\theta\bmod1$ on $h\mathbb N$, the sets $A_0$ and $B_0$ are $\phi^{-1}(I)$ and $\phi^{-1}(J)$ up to sets of density zero: up to density zero, $A$ and $B$ are lifts of parallel Bohr intervals from $h\mathbb N$, and $B$ contains almost all of the other residue classes. In case (2), the degenerate case, $B_1$ again covers $\mathbb N\setminus h\mathbb N$ up to density zero along the scale sequence, $B_0$ has density zero along it, and $A$ and $B$ are each invariant along it, up to density zero, under every shift by an element of $h\mathbb N$. The paper states that Theorem 1.4 resolves its Problem 1.5, the site's Problem 335, under the extra assumption that $B$ meets every residue class, and that the pairs of case (2) refute Erdős and Graham's speculation that every such pair arises from a rotation construction; Propositions 15.1 and 15.2 construct pairs in case (2), in Proposition 15.1 with $B$ of density zero.

Covers. The pairs with $d(A)>0$ and $d(A)+d(B)<1$ in which one of the sets meets every residue class: for them the theorem determines the structure the problem asks for. Not covered: pairs in which neither set meets every residue class, where Example 1.6 of the paper, a random subset $A$ of the even numbers with $d(A)=1/4$ and $d(A+A)=1/2$ almost surely, shows structure outside the Bohr description; and pairs with $d(A)+d(B)=1$, which the theorem's hypothesis excludes.

Depends on. No page of this wiki; the claim rests on the cited preprint.

Standing. Claimed. The paper is a preprint: its arXiv record carries one version, of 2026-04-14, and no journal reference. The site's curator, T. F. Bloom, records in the problem page's commentary that the problem is partially resolved by this paper under the residue-class assumption, but the site labels the problem OPEN (page last edited 2026-04-15), so the commentary is not acceptance and no reviewed evidence is listed. The proof is not checked here.