• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1515-1528
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• 2019

We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A{\subseteq}F^d_q$ and edges assigned the algebraic distance between pairs of vertices. We prove nontrivial results on locating specified subgraphs of maximum vertex degree at most t in dimensions $d{\geq}2t$.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.115-126
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• 2016

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.