Additive decompositions of large multiplicative subgroups in finite fields

Abstract

We show that a large multiplicative subgroup of a finite field ${\mathbb{F}}_q$ cannot be decomposed into $A+A$ or $A+B+C$ nontrivially. We also find new families of multiplicative subgroups that cannot be decomposed as the sum of two sets nontrivially. In particular, our results extensively generalize the results of Sárközy and Shkredov on the additive decomposition of the set of quadratic residues modulo a prime.

16 pages, revised based on referee comments

References in corpus (3)

• On the directions determined by Cartesian products and the clique number of generalized Paley graphs
• Quadratic residues and difference sets
• On sumsets involving $k$th powers of finite fields

Cited by in corpus (2)

• Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
• Restricted sumsets in multiplicative subgroups