Multiplicative Subgroups of Prime Fields Are Not Sumsets

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 Sárközy'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.

Disclosure

“nowledgements We thank Ben Hobson for interesting discussions during the early stages of this project while a Martingale scholar at Bristol. We also thank Bogdan Nica for informing us that Lemma 3.3 is a folklore identity of Sylvester. ChatGPT, accessed via a ChatGPT Plus subscription and using the GPT-5.4, GPT-5.5 and GPT-5.6 models, was used in the preparation of this manuscript for algebraic and numerical checks, generation of code, assistance with presentation, and proofread”

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2026-08-23T19:08:26.278867Z

Structural counts