Additive decompositions of large multiplicative subgroups in finite fields
Volume 213 / 2024
Acta Arithmetica 213 (2024), 97-116
MSC: Primary 11B30; Secondary 11P70, 11B13, 11T06
DOI: 10.4064/aa230330-21-2
Published online: 26 April 2024
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.
