Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric structures in pseudo-random graphs

Thang Pham, Steven Senger, Michael Tait, Vu Thi Huong Thu

TL;DR

This work develops a general framework for guaranteeing geometric configurations in pseudo-random graphs, translating finite-field geometric questions into graph-structural problems. Central to the approach are square-norms, a weak hypergraph regularity lemma, and a generalized von-Neumann estimate, which together yield three main configuration results: rectangles via Cartesian products, cycles with precise distribution in $\mathcal{G}=(n,d,\lambda)$, and disjoint copies of trees in large vertex sets. The authors provide two counting-lemma strategies for cycles—one via tensor products and one via direct expander methods—leading to improved thresholds and broad applicability beyond distance or dot-product graphs. These methods extend to diverse discrete-geometric questions and hint at extensions to other algebraic settings such as modules over finite rings.

Abstract

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

Geometric structures in pseudo-random graphs

TL;DR

This work develops a general framework for guaranteeing geometric configurations in pseudo-random graphs, translating finite-field geometric questions into graph-structural problems. Central to the approach are square-norms, a weak hypergraph regularity lemma, and a generalized von-Neumann estimate, which together yield three main configuration results: rectangles via Cartesian products, cycles with precise distribution in , and disjoint copies of trees in large vertex sets. The authors provide two counting-lemma strategies for cycles—one via tensor products and one via direct expander methods—leading to improved thresholds and broad applicability beyond distance or dot-product graphs. These methods extend to diverse discrete-geometric questions and hint at extensions to other algebraic settings such as modules over finite rings.

Abstract

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.
Paper Structure (20 sections, 19 theorems, 160 equations, 2 figures)

This paper contains 20 sections, 19 theorems, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cartesian product structures
  3. Distribution of cycles
  4. Distribution of disjoint trees
  5. Proof of Theorem \ref{['thm:CS']}
  6. Square-norm
  7. A weak hypergraph regularity lemma
  8. A generalized von-Neumann type estimate
  9. Proof of Theorem \ref{['thm:CS']}:
  10. Proof of Theorem \ref{['cycle-main1']}
  11. The first counting lemma for cycles
  12. The second counting lemma for cycles
  13. Distribution of paths
  14. Proof of Theorem \ref{['cycle-main1']} using the first counting lemma
  15. Proof of Theorem \ref{['cycle-main1']} using the second counting lemma
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $E$ be a set in $\mathbb{F}_q^d$. Assume that $|E|\gg q^{\frac{d+1}{2}}$, then

Figures (2)

  • Figure 1: Counting pairs of edges $xz$ and $yw$.
  • Figure 2: Counting edges $yz$.

Theorems & Definitions (31)

  • Theorem 1.1: Iosevich-Rudnev, EF3
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 21 more