Characterise positive-density sets with exactly additive sumset density: $d(A+B)=d(A)+d(B)$ (Erdős #335)
Statement
Let $d(A)$ denote the (asymptotic) density of $A\subseteq\mathbb{N}$, and $A+B=\{a+b : a\in A,\ b\in B\}$ the sumset. Characterise those $A,B\subseteq\mathbb{N}$ with positive density such that $$d(A+B)=d(A)+d(B).$$ One known mechanism: fix $\theta>0$ and sets $X_A,X_B\subseteq\mathbb{R}/\mathbb{Z}$ with $\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$, and take $A=\{n>0 : \{n\theta\}\in X_A\}$ and $B=\{n>0 : \{n\theta\}\in X_B\}$, where $\{x\}$ is the fractional part. Are all possible $A$ and $B$ generated in a similar way (using other groups)?
Acceptance. FULLY RESOLVES: a theorem, with complete proof, giving necessary and sufficient structural conditions describing ALL pairs $A,B\subseteq\mathbb{N}$ of positive density with $d(A+B)=d(A)+d(B)$ — stated in a precise form (e.g. group-rotation/level-set constructions on explicitly named compact groups, together with an explicit description of the residue-class-degenerate families within which membership is unconstrained), machine-checkable (Lean/Coq) preferred, otherwise a full written proof. ADVANCES: weakening or removing the Ackelsberg–Richter hypothesis that one set meets every residue class (e.g. a characterisation modulo an explicitly described degenerate part); a complete characterisation within a natural wider subfamily; or explicit new families of extremal pairs with a proof that they are not generated by the group-rotation mechanism in the statement (answering the 'are all examples of this form?' sub-question negatively). Deliver the proof file.
Background
Posed by Erdős and Graham [ErGr80, p.51]; listed as open on erdosproblems.com/335 (fetched 2026-07-13, status 'open', tagged 'number theory | additive combinatorics'). The condition $d(A+B)=d(A)+d(B)$ is the equality case of Kneser/Macbeath-type sumset inequalities for densities, and the rotation construction above transfers equality cases on the circle to $\mathbb{N}$. Major recent progress: Ackelsberg and Richter [AcRi26] resolved the problem under the additional hypothesis that one of the sets meets every residue class — roughly, if $d(A)>0$, $d(A+B)=d(A)+d(B)<1$, and $B$ meets every residue class, then either both sets arise from a rotation on the product of the circle with a finite cyclic group, or $A$ lies in a single residue class modulo some $h$ while $B$ looks like the union of all but one residue class modulo $h$. The site commentary cautions that a full characterisation with no such hypothesis 'seems hopeless': e.g. taking $A=B$ to be a random subset of the even integers of density $1/4$ gives $d(A+A)=1/2$, so degenerate residue-class phenomena admit essentially arbitrary noise. Any complete answer must therefore isolate and quantify the residue-class degeneracies. The attacker's tools: ergodic-theoretic and Kneser-type structure theory in the style of Ackelsberg–Richter — this is proof work with little computational purchase, though small-modulus case analysis can be machine-assisted.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #335 (T. F. Bloom) | website |
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