Additive Decompositions of Subgroups of Finite Fields

TL;DR

This paper investigates how multiplicative subgroups of finite fields can be expressed as sums of two sets, improving previous results on quadratic residues and primitive roots using advanced bounds and additive combinatorics.

Contribution

It provides new bounds and generalizations for additive decompositions of subgroups in finite fields, extending prior work on quadratic residues and primitive roots.

Findings

Improved bounds on additive decompositions of quadratic residues.

Generalizations of previous results on primitive roots.

Application of Karatsuba bound and additive combinatorics tools.

Abstract

We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalisations of results of C. Dartyge and A. Sarkozy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such the Karatsuba bound of double character sums and some results from additive combinatorics.