  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 $k\ge2$, we prove that
$$\lim_{x\rightarrow+\infty}\frac{B(x)}{\pi(x)}=0,$$ where $\pi(x)$ is the
number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set
$$\{p\le x: p\equiv1\pmod k; D_k\ \text{has a non-trivial 2-additive
decomposition}\}.$$
