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EMBEDDING DISTANCE GRAPHS IN FINITE FIELD VECTOR SPACES
Iosevich, Alex,Parshall, Hans Korean Mathematical Society 2019 대한수학회지 Vol.56 No.6
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 $d{\geq}2t$.
Embedding distance graphs in finite field vector spaces
Alex Iosevich,Hans Parshall 대한수학회 2019 대한수학회지 Vol.56 No.6
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 $A \subseteq \mathbf{F}_q^d$ 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 $d \geq 2t$.
VC-dimension and distance chains in $\mathbb{F}_q^d$
Ruben Ascoli,Livia Betti,Justin Cheigh,Alex Iosevich,Ryan Jeong,Xuyan Liu,Brian McDonald,Wyatt Milgrim,Steven J. Miller,Francisco Romero Acosta,Santiago Velazquez Iannuzzelli 강원경기수학회 2024 한국수학논문집 Vol.32 No.1
Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \F_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\Hh_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\Hh_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.
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.
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.