Sumsets and generalized arithmetic progressions in multiplicative subgroups

Abstract

Let $q=p^f$, and let $A\leq\mathbb{F}_q^\times$ be a multiplicative subgroup with $\mathbb{F}_p(A)=\mathbb{F}_q$. We prove that a proper subgroup $A$ is a generalized arithmetic progression (GAP) if and only if $|A| \in \{1, 2, 4\}$, and we determine when the full group $\mathbb{F}_q^\times$ is a GAP. For certain families of subgroups, we obtain the stronger conclusion that $A$ is additively irreducible. In particular, if $|A|>4$ and $p^e\equiv-1\pmod{|A|}$ for some $e\ge1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c \neq 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[\mathbb{F}_q^\times:A] \ge3$ and $|A| \ge 5$, which may be of independent interest.