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

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 $${\Delta }_k(E)=\{\Vert x^1\pm x^2\pm \cdots \pm x^k\Vert \in {\mathbb{F}}_q:x^j\in E,j=1,2,\dots ,k\},$$ where $\Vert \alpha \Vert ={\alpha }_1^2+\cdots +{\alpha }_d^2$ for $\alpha =({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{F}}_q^d.$ 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 ${\mathbb{F}}_q^{\ast }$. This problem is an extension of the Erdős-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.

References in corpus (3)

• Group actions and geometric combinatorics in ${\mathbb{F}}_q^d$
• Distance sets of two subsets of vector spaces over finite fields
• On the sums of any k points in finite fields