The $k$-resultant modulus set problem on algebraic varieties over finite fields

TL;DR

This paper investigates the minimal size of subsets within algebraic varieties over finite fields needed to ensure their $k$-resultant modulus set covers the entire field, extending the Erdős-Falconer distance problem.

Contribution

It introduces new bounds for the $k$-resultant modulus set problem when the set is contained in algebraic varieties over finite fields, utilizing energy estimates.

Findings

Established bounds for the size of sets on algebraic varieties

Extended the $k$-resultant modulus set problem to algebraic varieties

Used energy estimates to derive main results

Abstract

We study the $k$-resultant modulus set problem in the $d$-dimensional vector space $\mathbb F_q^d$ over the finite field $\mathbb F_q$ with $q$ elements. Given $E\subset \mathbb F_q^d$ and an integer $k\ge 2$, the $k$-resultant modulus set, denoted by $\Delta_k(E)$, is defined as $$ \Delta_k(E)=\{\|x^1\pm x^2 \pm \cdots \pm x^k\|\in \mathbb F_q: x^j\in E, ~j=1,2,\ldots, k\},$$ where $\|\alpha\|=\alpha_1^2+\cdots+ \alpha_d^2$ for $\alpha=(\alpha_1, \ldots, \alpha_d) \in \mathbb F_q^d.$ In this setting, the $k$-resultant modulus set problem is to determine the minimal cardinality of $E\subset \mathbb F_q^d$ such that $\Delta_k(E) = \mathbb F_q$ or $\mathbb{F}_q^*$. This problem is an extension of the Erd\H{o}s-Falconer distance problem. In particular, we investigate the $k$-resultant modulus set problem with the restriction that the set $E\subset \mathbb F_q^d$ is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.