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

TL;DR

This paper improves and extends results on the $k$-resultant modulus set problem over finite fields, demonstrating that under certain conditions, the set of sums of $k$ elements from a subset covers all nonzero elements of the field.

Contribution

The authors enhance previous bounds for the $k$-resultant modulus set problem on varieties over finite fields, broadening the applicability of the result.

Findings

Improved bounds for the size of subset A needed for the result.

Extended the result to more general varieties.

Confirmed the set covers all nonzero elements under new conditions.

Abstract

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.