Disjoint Sumsets Construction

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All groups here are finite abelian and written additively. For subsets $B,C$ of a group, $B+C=\{b+c:b\in B,c\in C\}$. Equality $|B+C|=|B||C|$ means that the addition map $B\times C\to B+C$ is injective. The asymptotic assertions below mean a sequence of constructions indexed by integers $n\to\infty$ along an unbounded subsequence. A quantity $n^{2+o(1)}$ has logarithm divided by $\log n$ tending to $2$.

Statement recorded in the supplied source.

Are there arbitrarily large integers $n$, finite abelian groups $H_n$, and subsets $A_1,\ldots,A_n,B_1,\ldots,B_n\subset H_n$ such that \[ |H_n|=n^{2+o(1)},\qquad \min_{1\leq i\leq n}|A_i||B_i|\geq n^{2-o(1)},\qquad |A_i+B_i|=|A_i||B_i|\quad(1\leq i\leq n), \] and \[ (A_i+B_i)\cap(A_j+B_k)=\varnothing \quad\text{for all }1\leq i,j,k\leq n\text{ with }j\ne k? \] More explicitly, for every $\epsilon>0$ and $M\in\mathbb N$, must some $n\geq M$ admit such subsets with $n^{2-\epsilon}\leq|H_n|\leq n^{2+\epsilon}$ and $|A_i||B_i|\geq n^{2-\epsilon}$ for every $i$, while the two exact addition conditions hold? [S2]

Short English statement

Can a nearly quadratic-size abelian group support arbitrarily many nearly quadratic-size injective addition pairs whose matched sums avoid every mismatched sum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 167 (GREEN-079). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Henry Cohn, Robert Kleinberg, Balázs Szegedy and Christopher Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379--388; arXiv:math/0511460v1, §4, Definition 4.1 and Conjecture 4.7. Replacing each source subset $B_i$ by its negative converts the difference-set convention into the displayed sum-set convention. https://arxiv.org/abs/math/0511460.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 36 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.