Sumsets and generalized arithmetic progressions in multiplicative subgroups

Abstract

Let $q=p^f$, and let $A\le {\mathbb{F}}_q^{\times }$ 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 $p^e\equiv -1(\mathrm{mod}|A|)$ for some $e\ge 1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c\ne 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[{\mathbb{F}}_q^{\times }:A]\ge 3$ and $|A|\ge 5$, which may be of independent interest.

16 pages

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