On sumsets involving $k$th powers of finite fields

In this paper, we study some topics concerning the additive decompositions of the set $D_k$ of all $k$th power residues modulo a prime $p$. For example, given a positive integer $k\ge 2$, we prove that $$\underset{x\to +\mathrm{\infty }}{\lim }\frac{B(x)}{\pi (x)}=0,$$ where $\pi (x)$ is the number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set $$\{p\le x:p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}k);D_k\text{ has a non-trivial 2-additive decomposition}\}\mathrm{.}$$