Related papers: A point-sphere incidence bound in odd dimensions a…

We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.

We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations…

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…

We considered the problem of obtaining upper bounds for the mathematical expectation of the $q$-norm ($2\leqslant q \leqslant \infty$) of the vector which is uniformly distributed on the unit Euclidean sphere. We finish the paper with…

Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…

We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through…

The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in $\mathbb{F}_q^2$. More precisely, for $A, B, C\subset \mathbb{F}_q^2$, if…

A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_{\epsilon}n^{1+\epsilon}$ times (for any $\epsilon>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the…

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums,…

Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of…

We consider the nonlinear Schr\"odinger equation$$-\Delta u + V(x)\,u = a\,u^p + \mu u \quad \text{in }\mathbb{R}^n,\qquad \int_{\mathbb{R}^n} u^2 = 1,$$modeling attractive Bose--Einstein condensates. For all dimensions $n\ge 2$ and all…

We prove that if $A \subset {\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the…

We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\log K_+ \leq c(d) n \log n$.

Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $<u,v >\notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0…

Given two points $p,q$ in the real plane, the signed area of the rectangle with the diagonal $[pq]$ equals the square of the Minkowski distance between the points $p,q$. We prove that $N>1$ points in the Minkowski plane $\R^{1,1}$ generate…

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for…

For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves…

Let $\mathcal{P}$ be a set of $m$ points and $\mathcal{L}$ a set of $n$ lines in $K^2$, where $K$ is a field with char$(K)=0$. We prove the incidence bound $$\mathcal{I}(\mathcal{P},\mathcal{L})=O(m^{2/3}n^{2/3}+m+n).$$ Moreover, this bound…

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss.…

We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…