Additive Decompositions of Subgroups of Finite Fields
Igor E. Shparlinski · SIAM Journal on Discrete Mathematics · 2013
We say that a set ${\mathcal S}$ is additively decomposed into two sets ${\mathcal A}$ and ${\mathcal B}$ if ${\mathcal S} = \{a+b:a\in {\mathcal A}, b \in {\mathcal B}\}$. Here we study additive decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalizations of results of Dartyge and Sárközy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such as the Karatsuba bound of double character sums and some results from additive combinatorics.