Dot-product graphs in finite fields

TL;DR

This work analyzes dot-product graphs in finite fields $\mathbb{F}_q^d$ by formulating the edge-labeling problem as the image $\Pi_G(S)$ of a graph-structured set $S$. Using finite-field Fourier analysis, it proves density results for connected graphs: if every adjacent pair of vertex-sets satisfies $|A_i||A_j|\ge C q^{d+k-1}$, then $|\Pi_G(S)|$ captures a positive proportion of $\mathbb{F}_q^{|E|}$ (Theorem ['tuxin']). For trees, it further shows that $\Pi_G(A_1\times\cdots\times A_{k+1})$ contains $(\mathbb{F}_q^*)^k$ under the same type of size conditions (Theorem ['shuxin']). The methods combine detailed Fourier-analytic bounds with combinatorial decompositions, extending prior results on dot-product configurations in finite fields and enhancing understanding of when geometric incidence graphs exhibit positive density. These results have potential implications for incidence geometry in finite fields and related harmonic-analysis approaches in discrete settings.

Abstract

In this paper, we study the dot-product graphs in $\mathbb{F}_q^d$. We prove that if the size of the product of two adjacent sets is large enough, then the set of dot-product graphs has positive density. Our method is based on finite field Fourier analytic techniques.

Theorems & Definitions (14)

• Theorem 1.1: MR2917133
• Theorem 1.2: MR4539819
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3