[Paper Review] Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness

Derrick Hart, Alex Iosevich+3 more|ArXiv.org|Apr 18, 2008

Limits and Structures in Graph Theory11 references6 citations

TL;DR

This paper investigates distance graphs in d-dimensional vector spaces over finite fields, establishing that the family of colored distance graphs $ G_q^ riangle $ is kaleidoscopically pseudo-random, meaning they exhibit uniform edge distribution and contain all finite configurations when large enough. The key result is a sharp estimate showing that sufficiently large subsets contain all $ k $-point configurations with specified distances, with applications to diameter bounds and pseudo-randomness in finite geometry.

ABSTRACT

In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations.

Motivation & Objective

• To systematically study the structural and pseudo-random properties of distance graphs in $ \mathbb{F}_q^d $, particularly focusing on coloring and configuration embedding.
• To establish that the family of colored distance graphs $ \{G_q^ riangle\} $ is kaleidoscopically pseudo-random, ensuring balanced edge distribution and asymptotic completeness.
• To prove that sufficiently large subsets of $ \mathbb{F}_q^d $ contain all possible finite $ k $-point configurations with prescribed distances, generalizing prior results.
• To compute the diameter of the distance graph $ G_q^ riangle $ in two and higher dimensions, showing it is three for $ q \geq 17 $, and never two for $ q \neq 3 $.
• To develop and apply character sum and Gauss sum techniques to estimate intersections of algebraic and non-algebraic varieties in $ \mathbb{F}_q^d $.

Proposed method

• Uses character sum methods and Gauss sum estimates to analyze the number of solutions to $ ||x - y|| = t $, leading to uniform distribution results.
• Applies orthogonality relations and additive character sums to bound the number of solutions to systems of quadratic equations defining configurations.
• Employs complete square completion and variable substitution in character sum expressions to derive lower bounds on solution counts.
• Introduces the concept of kaleidoscopic pseudo-randomness, requiring balanced edge distribution, asymptotic completeness, and universal configuration containment.
• Leverages results from classical Gauss sums and quadratic character sums to estimate the number of solutions to $ T^*(s,t,c) = 0 $, crucial for diameter analysis.
• Uses case analysis based on field characteristics and parameters to handle degeneracies in the solution space, particularly for $ q = 3, 5, 9, 13 $.

Experimental results

Research questions

• RQ1 Does the family of colored distance graphs $ G_q^ riangle $ in $ \mathbb{F}_q^d $ exhibit kaleidoscopic pseudo-randomness, defined by balanced edge distribution and universal configuration containment?
• RQ2 What is the diameter of the distance graph $ G_q^ riangle $ in two dimensions, and why is it never two for $ q \neq 3 $?
• RQ3 Under what conditions does a subset $ E \subset \mathbb{F}_q^d $ of size $ |E| \geq C q^{d(k-1)/k} q^{n/k} $ contain all $ k $-point $ J $-configurations with $ n $ prescribed distances?
• RQ4 How do Gauss sum estimates and character sum techniques enable precise counting of solutions to distance equations in finite fields?
• RQ5 Can the number of $ k $-point configurations with specified distances be estimated asymptotically as $ (1+o(1))|E|^k q^{-n} $, and under what conditions does this hold?

Key findings

• The family of colored distance graphs $ \{G_q^ riangle\} $ is kaleidoscopically pseudo-random, with edge color distribution satisfying $ |\mathcal{E}_j^i| \sim (1+o(1))|\mathcal{E}_j^{i'}| $ and asymptotic completeness as $ q \to \infty $.
• For any $ t \in \mathbb{F}_q $, the number of pairs $ (x,y) \in \mathbb{F}_q^d \times \mathbb{F}_q^d $ with $ ||x - y|| = t $ is $ (1+o(1))q^{2d-1} $, except when $ d=2 $ and $ t=0 $, where it is $ (2+o(1))q^{2d-1} $.
• If $ |E| \geq C q^{d(k-1)/k} q^{n/k} $ with sufficiently large $ C $, then the number of $ k $-point $ J $-configurations in $ E $ is $ |\mathcal{T}_k^J(E)| = (1+o(1))|E|^k q^{-n} $.
• The diameter of $ G_q^ riangle $ in two dimensions is three for $ q \geq 17 $, and it is never two for $ q \neq 3 $, as shown via character sum estimates and solution counting.
• The sum $ \sum_{t \in \mathbb{F}_q} \psi(g(t,c)) $, where $ \psi $ is the quadratic character, satisfies $ \left| \sum \psi(g(t,c)) \right| \leq 3q^{1/2} $, which rules out zero solution count for $ q \geq 17 $.
• The proof establishes that $ R(a,b,r) > 0 $ for $ q \geq 17 $, implying that any two points in $ S_r + a $ and $ S_r + b $ can be connected via a path of length three, confirming diameter three.