Exact subgroup self-sumsets and a robust quotient-character leakage bound

Definitions and mathematical statement

For an odd prime $p$, let $\mathbb F_p^*$ be the multiplicative group of its nonzero elements. A proper multiplicative subgroup $H<\mathbb F_p^*$ is required to differ from the whole multiplicative group. For $A\subseteq\mathbb F_p$, write $A+A=\{a+b:a,b\in A\}$, allowing equal summands. The containment $A+A\subseteq H$ includes every such sum, so in particular it excludes zero from $A+A$.

Define \[ Q(p)=\max\{|A|:A\subseteq\mathbb F_p, \exists H<\mathbb F_p^*\text{ with }A+A\subseteq H\}. \] The maximum includes the empty set and is therefore always defined. The explicit containment version of the workshop question asks whether \[ Q(p)=p^{o(1)}\qquad(p\to\infty\text{ through odd primes}), \] meaning that for every $\varepsilon>0$ there is $p_0(\varepsilon)$ such that $Q(p)\leq p^\varepsilon$ for all primes $p\geq p_0$. Equivalently, can a fixed positive power of $p$ ever be exceeded along infinitely many primes by a set whose every pairwise sum lies in one proper multiplicative subgroup? A negative answer to the proposed subpower bound requires such an infinite family for some fixed exponent. The subgroup may depend on the prime and on $A$; uniformity over all proper subgroups is part of the assertion. The source also asks about exact equality $A+A=H$. The classification in [S3] rules out that equality for $|A|>1$, while the containment target formulated here is a different remaining question.

Short English statement

Can a set of polynomial size in a prime field have every pairwise sum inside one proper multiplicative subgroup?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001138 (AIM-COMBINATORICS-0263), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 4.7. Original workshop question underlying AIM-COMBINATORICS-0263. https://aimath.org/WWN/additivecomb/additivecomb.pdf.
• S3 Chi Hoi Yip and Semin Yoo, Additive decompositions of multiplicative subgroups via differential identities, Theorem 1.1. Resolution of the separate exact-equality question. https://arxiv.org/pdf/2608.02568v2.