Erdos distance problem in vector spaces over finite fields

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $|\Delta(E)| \gtrsim q$ whenever $|E| \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. The second listed author and Misha Rudnev established the threshold $\frac{d+1}{2}$, and the authors of t...

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $\Delta(A)$ contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$ in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture ...

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $\Delta(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-p...

We study a matrix analog of the Erd\H{o}s-Falconer distance problems in vector spaces over finite fields. There arises an interesting analysis of certain quadratic matrix Gauss sums.

We study the following two-parameter variant of the Erd\H os-Falconer distance problem. Given $E,F \subset {\Bbb F}_q^{k+l}$, $l \geq k \ge 2$, the $k+l$-dimensional vector space over the finite field with $q$ elements, let $B_{k,l}(E,F)$ be given by $$\{(\Vert x'-y'\Vert, \Vert x"-y" \Vert): x=(x',x") \in E, y=(y',y") \in F; x',y' \in {\Bbb F}_q^k, x",y" \in {\Bbb F}_q^l \}.$$ We prove that if $|E||F| \geq C q^{k+2l+1}$, then $B_{k,l}(E,F)={\Bbb F}_q \times {\Bbb F}_q$. Fu...

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016). More precisely, we prove that for $\mathcal{E}\subseteq \mathbb{F}_q^d$, if $d=2$ and $q^{1+\frac{1}{4k-1}}=o(|\mathcal{E}|)$ then we have $|k\Delta_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q$; if $d\ge 3$ and $q^{\frac{d}{2}+\frac{1}{2k}}=o(|\m...

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of distances between the point y and the set E is similar to the size of the finite field \mathbb F_q. As a corollary, we obtain that for each set E\subseteq \mathbb F_q^d with |E|\ge q, there exists a set Y\subseteq \mathbb F_q^d with |Y|\sim q...

We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of...

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in E^4$ such that $ Q(x-y)/Q(z-w)=r,$ where $Q$ is a non-degenerate quadratic form in $d$ variables over $\mathbb F_q.$ When $Q(\alpha)=\sum_{i=1}^d \alpha_i^2$ with $\alpha=(\alpha_1, \ldots, \alpha_d)\in \mathbb F_q^d,$ Pham (2022) recently u...