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LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS
BENNETT, MICHAEL,CHAPMAN, JEREMY,COVERT, DAVID,HART, DERRICK,IOSEVICH, ALEX,PAKIANATHAN, JONATHAN Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
Let $E{\subset}{\mathbb{F}}^d_q$, the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y 2 E by an edge if ${\parallel}x-y{\parallel}:=(x_1-y_1)^2+{\cdots}+(x_d-y_d)^2=1$. We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1{\cdot}{\mid}E{\mid}^{k+1}q^{-k}$ plus a much smaller remainder.
Long paths in the distance graph over large subsets of vector spaces over finite fields
Michael Bennett,Jeremy Chapman,David Covert,Derrick Hart,Alex Iosevich,Jonathan Pakianathan 대한수학회 2016 대한수학회지 Vol.53 No.1
Let $E \subset {\mathbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||:={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that the non-overlapping chains of length $k$, with $k$ in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1 \cdot {|E|}^{k+1}q^{-k}$ plus a much smaller remainder.