Related papers: On the Erd\H{o}s-Falconer distance problem for two…
In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane…
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 Erd\"os/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to…
art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\"os-Falconer distance conjecture holds for subsets of the unit sphere in $\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this…
In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…
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 $B_{k,l}(E,F)$…
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…
In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular,…
In this paper, we prove Erd\H{o}s distance conjecture in $\mathbb{R}^d$, namely, a set of $n$ points in $\mathbb{R}^2$ determines $\Omega(\frac{n}{\sqrt{\log n}})$ distances, and for $d\ge 3$, a set of $n$ points in $\mathbb{R}^d$…
We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances.…
We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.
We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of…
Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $\Delta_{d, d}(E)=\left\{\left(\|x_1-y_1\|,…
We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$…
This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…
We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…
We prove a series of results on the size of distance sets corresponding to sets in the Euclidean space. These distances are generated by bounded convex sets and the results depend explicitly on the geometry of these sets. We also use a…
In this paper, we study the distance problem in the setting of finite p-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarify the conjecture and provide examples to support it. Surprisingly,…
Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg…
We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections…