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Given $E subseteq mathbb{F}_q^d times mathbb{F}_q^d$, with the finite field $mathbb{F}_q$ of order $q$ and the integer $d ge 2$, we define the two-parameter distance set as $Delta_{d, d}(E)=left{left(|x_1-y_1|, |x_2-y_2|right) : (x_1,x_2), (y_1,y_2) in E right}$. Birklbauer and Iosevich (2017) proved that if $|E| gg q^{frac{3d+1}{2}}$, then $ |Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| gg q^{frac{10}{3}}$, then $ |Delta_{2, 2}(E)| gg q^2$. In this paper, we present extensions and improvements of these results.
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work
In this paper we obtain a new lower bound on the ErdH{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $Asubset mathbb{F}_p^2$ with $|A|le p^{7/6}$, the number of distinct distances determined by
The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $lfloor n^2/4 rfloor +t$ edges and triangle covering number $s$, we determine (for large
Given a sequence $mathbf{k} := (k_1,ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ conta
Let $textbf{k} := (k_1,ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ con