The Mattila-Sj\"olin problem for the k-distance over a finite field

Abstract

Let $\mathbb{F}_q^d$ be a $d$-dimensional vector space over a finite field $\mathbb{F}_q$ with $q$ elements. For $x\in \mathbb{F}_q^d$, let $\|x\| = x_1^2+\dots+x_d^2$. By abuse of terminology, we shall call $\|\cdot\|$ a norm on $\mathbb{F}_q^d$. For a subset $E\subset \mathbb{F}_q^d$, let $\Delta(E)$ be the distance set on $E$ defined as $\Delta(E):=\{\|x-y\| : x, y \in E \}$. The Mattila-Sj\"olin problem seeks the smallest exponent $\alpha>0$ such that $\Delta(E) =\mathbb{F}_q$ for all subsets $E \subset \mathbb{F}_q^d$ with $|E| \geq Cq^\alpha$. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm $\|\cdot\|.$ Namely, we replace the norm $\|\cdot\|$ by the so-called $k$-norm $(1 \leq k \leq d)$, which can be viewed as a kind of deformation of $\|\cdot\|$. To derive our result on the Mattila-Sj\"olin problem for the $k$-norm, we use a combinatorial method to analyze various summations arising from the discrete Fourier machinery. Even though our distance set is smaller than the one in the Mattila-Sj\"olin problem, for some $k$ we still obtain the same result as that of Iosevich and Rudnev (2007), which deals with the Mattila-Sj\"olin problem. Furthermore, our result is sharp in all odd dimensions.