Threshold functions for substructures in random subsets of finite vector spaces

TL;DR

This paper develops a general framework to analyze the emergence of specific substructures in random subsets of finite vector spaces, revealing threshold behaviors and distribution limits for various configurations.

Contribution

It introduces a novel framework for threshold analysis in finite vector spaces and applies it to multiple combinatorial structures, including arithmetic progressions and geometric configurations.

Findings

Established coarse threshold results for various substructures.

Proved a limiting Poisson distribution at the threshold scale.

Identified thresholds for sum-free sets and Sidon sets in finite vector spaces.

Abstract

The study of substructures in random objects has a long history, beginning with Erd\H{o}s and R\'enyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.