Related papers: Distribution of distances in positive characterist…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

For a set of distances D={d_1,...,d_k} a set A is called D-avoiding if no pair of points of A is at distance d_i for some i. We show that the density of A is exponentially small in k provided the ratios d_1/d_2, d_2/d_3, ..., d_{k-1}/d_k…

We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov-Galvin-Starchenko, this yields Szemer\'edi-Trotter-like incidence bounds for function fields over finite fields. We deduce…

Let $\mathcal{O}_K$ be the ring of integers of an algebraic number field $K$ embedded into $\mathbb{C}$. Let $X$ be a subset of the Euclidean space $\mathbb{R}^d$, and $D(X)$ be the set of the squared distances of two distinct points in…

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be…

We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot…

In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $\Omega\left(n^{7/9}/{\rm polylog}…

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…

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…

For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \Delta_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$…

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…

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…

Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $\Omega(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive…

The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the…

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erd\H{o}s in 1946 and rather recently proved by Guth and Katz. We show that when near…

Let $\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is…

There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…