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In this article, we consider the $c$-dispersion problem in a metric space $(X,d)$. Let $P=\{p_{1}, p_{2}, \ldots, p_{n}\}$ be a set of $n$ points in a metric space $(X,d)$. For each point $p \in P$ and $S \subseteq P$, we define…

We study the saddlepoint approximation (SPA) for sums of $n$ i.i.d. random vectors $X_i\in\mathbb R^d$ in growing dimensions. SPA provides highly accurate approximations to probability densities and distribution functions via the moment…

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon,…

A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set…

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least…

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set…

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a…

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

The Johnson-Lindenstrauss Lemma allows for the projection of $n$ points in $p-$dimensional Euclidean space onto a $k-$dimensional Euclidean space, with $k \ge \frac{24\ln \emph{n}}{3\epsilon^2-2\epsilon^3}$, so that the pairwise distances…

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum…

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 $…

Lightness and sparsity are two natural parameters for Euclidean $(1+\varepsilon)$-spanners. Classical results show that, when the dimension $d\in \mathbb{N}$ and $\varepsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits…

This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are…

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $|x_i - x_j| \leq 1$ for all $1 \leq i,j \leq n$. We show that the ratio of the number of ``neighbors''…

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…

Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that, for every positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$…

We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$-\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f $$ on a bounded open subset $\Omega\subset…

Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and…