On the $k$-resultant modulus set problem on varieties over finite fields

Let $V\subset {\mathbb{F}}_q^d$ be a $\mathit{\text{regular}}$ variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then we have $${\mathrm{\Delta }}_k(A)=\{\mathrm{\mid }{x}_1+\cdots +x_k\mathrm{\mid }\text{\hspace{0.17em}⁣}:x_i\in A\}\supseteq {\mathbb{F}}_q^{\ast }\mathrm{.}$$ In this paper, we provide improvements and extensions of their result.