Related papers: On incidences of lines in regular complexes

We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines…

We consider random lines in $\mathbb{R}^3$ (random with respect to the kinematic measure) and how they intersect $\mathbb{S}^2$. It is known that the entry point and the exit point behave like \textit{independent} uniformly distributed…

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…

A system of linear equations $L$ over $\mathbb{F}_q$ is common if the number of monochromatic solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$ is asymptotically at least the expected number of monochromatic solutions in a random…

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is…

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular…

We develop a new `subspace layered least squares' interior point method (IPM) for solving linear programs. Applied to an $n$-variable linear program in standard form, the iteration complexity of our IPM is up to an $O(n^{1.5} \log n)$…

Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is an odd prime power. A quadratic subspace $(W,Q)$ of $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ is called dot$_{k}$-subspace if $Q$ is isometrically…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict to the…

A finite \emph{$k$-net} of order $n$ is an incidence structure consisting of $k\ge 3$ pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one…

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$.

We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks)…

In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is…

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of…

We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular $(u,v)$-Furstenberg sets, which is new for large $u$ and $v$.

In this paper we study the number of incidences between $m$ points and $n$ varieties in $\mathbb{F}^d$, where $\mathbb{F}$ is an arbitrary field, assuming the incidence graph contains no copy of $K_{s,s}$. We also consider the analogous…

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees…