Additive Decompositions of Subgroups of Finite Fields

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.