[Paper Review] Embedding distance graphs in finite field vector spaces

Alex Iosevich, Hans Parshall|arXiv (Cornell University)|Feb 18, 2018

Finite Group Theory Research10 references3 citations

TL;DR

This paper establishes that large subsets of finite field vector spaces $[f_q^d$ contain isometric copies of any distance graph with $n$ vertices and maximum degree $t$, provided the dimension satisfies $d \geq 2t$ and the set size exceeds $12n^2 q^{(d-1)/2 + t}$. The result relies on Fourier analytic techniques and averaging arguments to show that such configurations are not only present but abundant, extending results on distance sets and point configurations in finite geometry.

ABSTRACT

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 $d \geq 2t$.

Motivation & Objective

• To determine under what conditions large subsets of $f_q^d$ contain isometric copies of arbitrary distance graphs with bounded maximum vertex degree.
• To extend finite field analogues of Euclidean distance problems, particularly those related to Falconer's conjecture and Erdős-type distance problems.
• To show that for $d \geq 2t$, sufficiently large sets in $f_q^d$ contain all distance graphs of size $n$ and degree $t$, even when $n$ is large.
• To establish a quantitative lower bound on the number of genuine, vertex-disjoint isometric copies of such graphs in large sets.

Proposed method

• Uses Fourier analytic methods to estimate the number of isometric copies of a distance graph $\mathcal{G}$ in a subset $A \subseteq \u0066_q^d$, relying on exponential sums and character sums.
• Applies a recursive averaging argument via induction on the number of edges, reducing the count of full configurations to subgraphs with one less edge.
• Introduces restricted averages $\mathbf{E}^{*}$ to isolate genuine configurations with distinct vertices, excluding degenerate cases where vertices coincide.
• Employs Cauchy-Schwarz and $L^2$-norm estimates to bound error terms arising from coincident vertex contributions in the counting process.
• Uses the structure of the distance graph and the maximum degree $t$ to control the number of overlapping configurations and bound the error in the main term.
• Relies on a key estimate (Theorem 6) that bounds the deviation of the expected number of configurations from the main term by $O(q^{t - (d+1)/2})$, which is small when $d$ is large relative to $t$.

Experimental results

Research questions

• RQ1 Under what conditions does a large subset $A \subseteq \u0066_q^d$ contain an isometric copy of any given distance graph with $n$ vertices and maximum degree $t$?
• RQ2 Can the dimension $d$ be chosen such that $d \geq 2t$ to guarantee the existence of all such distance graphs in large sets?
• RQ3 How many genuine, vertex-disjoint isometric copies of a fixed distance graph can be found in a large subset of $\u0066_q^d$?
• RQ4 What is the quantitative lower bound on the number of such configurations, and how does it depend on $n$, $t$, $q$, and $d$?
• RQ5 Can the error introduced by coincident vertices in the counting process be controlled to ensure the existence of non-degenerate configurations?

Key findings

• For any distance graph $\mathcal{G}$ with $n$ vertices, $m$ edges, and maximum vertex degree $t$, if $|A| \geq 12n^2 q^{(d-1)/2 + t}$, then $A$ contains at least $\frac{1}{2}|A|^n q^{-m}$ genuine isometric copies of $\mathcal{G}$.
• The number of isometric copies of $\mathcal{G}$ in $A$ is asymptotically close to $\alpha^n$, where $\alpha = |A|/q^d$, with error bounded by $O(n^2 \alpha^{n-1} q^{t - (d+1)/2})$.
• When $d \geq 2t$, the error term becomes subdominant to the main term, ensuring the existence of such configurations for sufficiently large $q$ and $n$.
• The method guarantees the existence of arbitrarily long cycle graphs (e.g., $C_n$) in $\u0066_q^d$ for $d \geq 4$, which is a new result in the finite field setting.
• The bound on the number of genuine configurations is robust: even after subtracting contributions from degenerate cases (coincident vertices), at least half of the expected configurations remain non-degenerate.
• The result extends to any non-degenerate quadratic form in place of the standard squared Euclidean distance, via known results from [10] and [2], showing the method is generalizable beyond the Euclidean model.