Journal of the
Korean Mathematical Society JKMS

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.

Keywords: Erdos distance problem, finite fields, graph theory

MSC numbers: 52C10, 11T23