Problem 335

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claims/: The 1 claim page of Problem 335, one per claimant's result; the problem's standing derives from them.

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Statement. Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that

$$d(A+B)=d(A)+d(B).$$

Status. Open, the site's label (OPEN; page last edited 2026-04-15). One partial claim is recorded: [[problems/additive_combinatorics/E0335/claims/2026_04_14_ackelsberg_richter|Ackelsberg and Richter's inverse theorem]], a preprint of 2026-04-14 whose Theorem 1.4 characterizes the pairs in which one of the sets meets every residue class; the characterization without that assumption is not settled.

Source. erdosproblems.com/335, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #335, https://www.erdosproblems.com/335.

References.

• [AcRi26] E. Ackelsberg and F. K. Richter, An inverse theorem for sumsets of sets of positive density in the integers. arXiv:2604.12864 (2026).

Formalization. None recorded.

Current assessment

The site's formulation above asks for the pairs $A,B\subseteq\mathbb N$ of positive density with $d(A+B)=d(A)+d(B)$. The site's commentary, in the corpus's words: equality holds when there is a $\theta>0$ and sets $X_A,X_B\subseteq\mathbb R/\mathbb Z$ with $\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$ such that $A$ and $B$ are the positive integers $n$ whose fractional parts $\{n\theta\}$ fall in $X_A$ and $X_B$, and the site asks whether every pair arises in a similar way from other groups. The site credits Ackelsberg and Richter [AcRi26] with a partial resolution under the assumption that one of the sets meets every residue class, recorded at [[problems/additive_combinatorics/E0335/claims/2026_04_14_ackelsberg_richter|Ackelsberg and Richter's inverse theorem]]: for $d(A)>0$, $d(A)+d(B)<1$ and $B$ meeting every residue class, either both sets come, up to density zero, from a rotation on a subsemigroup $h\mathbb N$ (lifts of parallel Bohr intervals), or $A$ lies in one residue class modulo some $h$ and $B$ covers almost all of the other classes, the paper's degenerate case. The site also records that a full characterization without that assumption looks hopeless, since a random subset $A$ of the even numbers of density $1/4$ has $d(A+A)=1/2$; the paper's Example 1.6 is this example. The claim is a preprint without outside review, so it stays claimed, and as a partial claim it leaves the problem open.

Search scope. The account above rests on the site's problem page, the arXiv record of the paper (version of 2026-04-14) and the library card of that version. The proof is not checked here, and no literature search beyond these sources is recorded.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• ackelsberg_2026_inverse_theorem_sumsets_sets_positive_density