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arXiv · 2607.24270

Multiplicative Subgroups of Prime Fields Are Not Sumsets

Misha Rudnev·Fred Tyrrell
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Abstract

Let $H \leq \mathbb{F}_p^*$ be a proper multiplicative subgroup, and suppose that $H = A+B$ for some $A,B \subseteq \mathbb{F}_p$. We prove that either one of the summands is a singleton, or $|A|=|B|=2$ and $|H|=4$. In particular, no proper multiplicative subgroup of $\mathbb{F}_p^*$ can be written as $A+B$ with $|A|,|B|>2$. Our proof builds on the Hanson-Petridis polynomial method and Kalmynin's subsequent resolution of S\'ark\"ozy's conjecture for quadratic residues. Using Kalmynin's $|A|=|B|$ theorem as a structural input, we develop uniform combinatorial and arithmetic arguments which apply to multiplicative subgroups of arbitrary index.

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Misha Rudnev, Fred Tyrrell. 2026-07-27. Multiplicative Subgroups of Prime Fields Are Not Sumsets. https://arxiv.org/abs/2607.24270

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