Related papers: An Improved Point-Line Incidence Bound Over Arbitr…

We solve a problem conjectured by Cheriyan, giving sharp bounds for incidence of certain edge partitions of the connected graph on $n$-vertices. We briefly discuss the history of the problem and relation to node connectivity of strongly…

We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\mathbb{F}$, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb{F})$ is positive, we have $$|A(A+1)| \gtrsim |A|^{11/9},…

In this note we prove an upper bound on the $\mathbb F_p$-rank of the incidence matrix of points and hyperplanes in $(\mathbb Z/p^k \mathbb Z)^n$, improving a recent bound of Laba and Trainer when $k$ is large.

We study configurations of $n$ points and $n$ lines that form $\Theta(n^{4/3})$ incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel…

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…

We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances…

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been…

As a variant of the celebrated Szemer\'edi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in $\mathbb{R}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}n^{3/4})$ incidences for…

We obtain some asymptotic formulae (with power savings in their error terms) for the number of quadruples in the Cartesian product of an arbitrary set $A \subset \mathbf{R}$ and for the number of quintuplets in $A\times A$ for any subset…

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy,…

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear…

For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from…

In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and…

In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the…

Guth and Katz proved that, as conjectured by Elekes and Sharir, $m$ lines in 3-space have at most constant times $ m^{3/2}$ intersection points, aside from some obvious counter examples. We give an explicit bound for the constant, using the…

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…

We prove essentially sharp incidence estimates for a collection of $\delta$-tubes and $\delta$-balls in the plane, where the $\delta$-tubes satisfy an $\alpha$-dimensional spacing condition and the $\delta$-balls satisfy a…

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main…

We study the problem of approximation of 2D set of points. Such type of problems always occur in physical experiments, econometrics, data analysis and other areas. The often problems of outliers or spikes usually make researchers to apply…

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…