The Mattila-Sjölin 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 $\Vert x\Vert =x_1^2+\cdots +x_d^2$. By abuse of terminology, we shall call $\Vert \cdot \Vert$ 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 $\Delta (E):=\{\Vert x-y\Vert :x,y\in E\}$. The Mattila-Sjölin problem seeks the smallest exponent $\alpha >0$ such that $\Delta (E)={\mathbb{F}}_q$ for all subsets $E\subset {\mathbb{F}}_q^d$ with $|E|\ge C{q}^{\alpha }$. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm $\Vert \cdot \Vert .$ Namely, we replace the norm $\Vert \cdot \Vert$ by the so-called $k$-norm $(1\le k\le d)$, which can be viewed as a kind of deformation of $\Vert \cdot \Vert$. To derive our result on the Mattila-Sjölin 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-Sjölin problem, for some $k$ we still obtain the same result as that of Iosevich and Rudnev (2007), which deals with the Mattila-Sjölin problem. Furthermore, our result is sharp in all odd dimensions.

16 pages. To be published in the Journal of the Korean Mathematical Society