Related papers: Point sets avoiding near-integer distances

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the…

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

Let $S = \{ {A_1},{A_2}, \cdots ,{A_n}\} $ be a finite point set in m-dimensional Euclidean space ${E^m}$, and$\left\| {{A_i}{A_j}} \right\|$ be the distance between $A_i$ and $A_j$. Define $\sigma (S) = \sum\limits_{1 \le i < j \le n}…

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…

We prove that for any $2<p<\infty$ and for every $n$-dimensional subspace $X$ of $L_p$, represented on $\mathbb R^n$, whose unit ball $B_X$ is in Lewis' position one has the following two-level Gaussian concentration inequality: \[ \mathbb…

We show that for every $\alpha > 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $\alpha$, but the whole space requires distortion at least $\Omega(\sqrt{\alpha \log…

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…

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''…

The following generalisation of the Erd\H{o}s unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given $k$ positive real numbers $\delta_1,\dots,\delta_k$, a $(k+1)$-tuple $(p_1,\dots,p_{k+1})$ in $\mathbb{R}^d$ is…

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains…

We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $\Omega\left(n^{32/39-\varepsilon}\right)$ distinct distances, for…

We revisit the classical Unit Distance Problem posed by Erd\H{o}s in 1946. While the upper bound of $O(n^{4/3})$ established by Spencer, Szemer'edi, and Trotter (1984) is tight for systems of pseudo-circles, it fails to account for the…

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{\alpha, -\alpha\}$,…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove…

Erd\H{o}s asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime…

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain…