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

Let $V\subset \mathbb{F}_q^d$ be a \textit{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 $$\Delta_{k}(A)=\{|x_1+\cdots+x_k|\colon x_i\in A\}\supseteq \mathbb{F}_q^*.$$ In this paper, we provide improvements and extensions of their result.