Multiplicative Subgroups of Prime Fields Are Not Sumsets

Abstract

Let $H\le {\mathbb{F}}_p^{\ast }$ 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 ${\mathbb{F}}_p^{\ast }$ 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.

41 pages