J. Korean Math. Soc. 2016; 53(1): 115-126
Printed January 1, 2016
https://doi.org/10.4134/JKMS.2016.53.1.115
Copyright © The Korean Mathematical Society.
Michael Bennett, Jeremy Chapman, David Covert, Derrick Hart, Alex Iosevich, and Jonathan Pakianathan
Rochester Institute of Technology, Lyon College, University of Missouri-Saint Louis, Rockhurst University, University of Rochester, University of Rochester
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
2019; 56(6): 1515-1528
2023; 60(4): 799-822
2019; 56(1): 113-125
2015; 52(1): 209-224
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd