相关论文: Many unit distances requires many directions

Any $n$-tuple of points in the plane can be moved to any other $n$-tuple by a continuous motion with at most $\binom{n}{3}$ intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there…

Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some…

We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}^D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n^{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This…

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 $\mathbb{R}^d$…

It is proved that for any mapping of a unit segment to a unit square, there is a pair of points of the segment for which the square of the Euclidean distance between their images exceeds the distance between them on the segment by at least…

A recent generalization of the Erd\H{o}s Unit Distance Problem, proposed by Palsson, Senger and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in $3$-space. Studying a variant of…

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…

In a previous paper we showed that, for any $n \ge m+2$, most sets of $n$ points in $\RR^m$ are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there…

We consider the directed Hausdorff distance between point sets in the plane, where one or both point sets consist of imprecise points. An imprecise point is modelled by a disc given by its centre and a radius. The actual position of an…

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…

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…

We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a…

We study the minimum number of distinct distances between point sets on two curves in $R^3$. Assume that one curve contains $m$ points and the other $n$ points. Our main results: (a) When the curves are conic sections, we characterize all…

We compute the distance-dependent three-point function of general planar maps and of bipartite planar maps, i.e., the generating function of these maps with three marked vertices at prescribed pairwise distances. Explicit expressions are…

This paper formulates and evaluates a series of multi-unit measures of directional association, building on the pairwise {\Delta}P measure, that are able to quantify association in sequences of varying length and type of representation.…

We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal P,$ a set of $n$ points, and a sequence of distances $(\delta_1,\ldots,\delta_k)$, we study the maximum possible number of tuples of…

The Szemer\'edi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an…

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets:…

Guth and Katz proved that any point set $\mathcal P$ in the plane determines $\Omega(|\mathcal P|/\log|\mathcal P|)$ distinct distances. We show that when near to this lower bound, a point set $\mathcal P$ of the form $A\times A$ must…

We prove a special case of Erd\H{o}s' unit distance problem using a corollary of the subspace theorem bounding the number of solutions of linear equations from a multiplicative group. We restrict our attention to unit distances coming from…