Related papers: Distinct distances between points and lines

We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of…

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 show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…

For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha,$ $ \cdots, (N-1) \alpha$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one…

It is shown that given a set of $N$ points in the plane or on the sphere, there is a subset of size $\gtrsim N^{1/3}/\log N$ with all pairwise distances between points distinct.

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the…

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…

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly…

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…

In this note, we show that in planar pointsets determining many unit distances, these unit distances must span many directions. Specifically, we show that a set of $n$ points can determine only $o(n^{4/3})$ unit distances from a set of at…

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz \cite{GK}, to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum…

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 establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…

We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two…

In this paper we obtain a new lower bound on the Erd\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances…

We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no…

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in…

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…