The Mattila-Sjölin problem for the k-distance over a finite field

TL;DR

This work extends the Mattila-Sjölin framework to a deformed distance, the $k$-norm, over finite fields. Using a Fourier-analytic approach and a careful decomposition of the sphere transform into $A(m,t)$ and $B(m)$ components, the authors establish that for odd $d$ and any $k$, if $|E|\ge Cq^{\max\{(d+1)/2,\ d-k\}}$, then the $k$-distance set $D_k(E)$ equals the whole field $\mathbb{F}_q$. In even dimensions a similar bound is obtained, matching the original MS threshold in many regimes and proving sharpness in odd $d$. The results demonstrate robustness of the MS-type phenomenon under a natural deformation of distance and introduce a new combinatorial elimination technique to control intricate sum terms. The work advances discrete distance problems in finite fields and highlights the power of Fourier-analytic methods combined with careful combinatorial analysis.

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 $Δ(E)$ be the distance set on $E$ defined as $Δ(E):=\{\|x-y\| : x, y \in E \}$. The Mattila-Sjölin problem seeks the smallest exponent $α>0$ such that $Δ(E) =\mathbb{F}_q$ for all subsets $E \subset \mathbb{F}_q^d$ with $|E| \geq Cq^α$. 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-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.

Theorems & Definitions (17)

• Conjecture 1.3
• Theorem 1.4
• Proposition 2.1: Orthogonality of characters
• proof
• Definition 2.2: Gauss sum
• Proposition 2.3: Theorem 5.15, LN97
• Lemma 2.4
• proof
• Definition 2.5
• Proposition 2.6