Related papers: The Erd\H{o}s-Falconer distance problem between ar…

There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality $s$. In this paper…

For $ E\subset \mathbb{F}_q^d$, let $\Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $…

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for…

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points,…

The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect…

A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distinct distances between two distinct points in $X$ and a subset $X$ is called a locally $k$-distance set if for any point $x$ in…

Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle,…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…

A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates…

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set $S$ of $s$ points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in…

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously…

The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq…

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…

Given a set $X\subseteq\mathbb{R}^2$ of $n$ points and a distance $d>0$, the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$. Let $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ denote the…

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…

Erd\H{o}s and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erd\H{o}s distinct distances problem. We consider the analogous problem for…

We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles…

A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with the property that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Ellenberg and Erman proved that…