LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS

Title & Authors
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;

Abstract
Let $\small{E{\subset}{\mathbb{F}}^d_q}$, 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 $\small{{\parallel}x-y{\parallel}:=(x_1-y_1)^2+{\cdots}+(x_d-y_d)^2=1}$. 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, $\small{1{\cdot}{\mid}E{\mid}^{k+1}q^{-k}}$ plus a much smaller remainder.
Keywords
Erdos distance problem;finite fields;graph theory;
Language
English
Cited by
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