For a fixed integer k≥ 2, let Gamma be a finite abelian group of order n with no nonzero element whose order divides k. Define f_k(Gamma) to be the largest integer such that every subset B⊆ Gamma satisfying |B|≥ n-f_k(Gamma) can be written as a k-fold sumset B=A+…+A for some A⊆Gamma. The conjecture is that [ f_k(Gamma)=widetildeTheta!(n^1/k) ] as n→∞ (with k fixed).

k-fold sumset threshold conjecture

For a fixed integer $k\ge 2$ , let $\Gamma$ be a finite abelian group of order $n$ with no nonzero element whose order divides $k$ . Define $f_k(\Gamma)$ to be the largest integer such that every subset $B\subseteq \Gamma$ satisfying $|B|\ge n-f_k(\Gamma)$ can be written as a $k$ -fold sumset $B=A+\cdots+A$ for some $A\subseteq\Gamma$ . The conjecture is that

f_k(\Gamma)=\widetilde{\Theta}\!\left(n^{1/k}\right)

as $n\to\infty$ (with $k$ fixed).

References

Primary source

arXiv

Additional references

Random Cayley sum hypergraphs and k-fold sumsets — arXiv — Jihyo Chae, Hyunwoo Lee

Posted 13 hours ago · edited 12 hours ago

Progress summary

Refreshed 12 hours ago

Claimed progress

A new preprint narrows the gap between the best known upper and lower bounds, but does not prove the conjectured answer in general.

The k-fold sumset threshold conjecture predicts a matching-order threshold for dense subsets that are not k-fold sumsets. Its general case remains unresolved.

October 2026 improved bounds

Jihyo Chae and Hyunwoo Lee claim an upper bound of roughly $n^{(2k-1)/(4k-3)}$ and a lower bound of roughly $n^{1/k}$ for groups with no nontrivial element whose order divides $k$ . The upper estimate improves the general bound and recovers the best known case for $k=2$ , but the conjectured matching order is not proved in general.

Current status (as of October 2026): A preprint claims improved upper and lower bounds, while the conjectured matching order remains open in general.

Sources

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