Threshold functions for incidence properties in finite vector spaces

TL;DR

This paper establishes threshold functions for various incidence properties in finite vector spaces, advancing understanding of random point sets and their geometric configurations.

Contribution

It provides new threshold functions for incidence events in finite vector spaces, extending recent results and offering insights into random geometric configurations.

Findings

Derived threshold functions for point-flat incidences

Extended Chen and Greenhill's recent results

Analyzed properties like m-blocking sets and incidences

Abstract

The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that there is an $\ell$-rich $m$-flat with regard to a random set of points in $\mathbb{F}_q^n$, the event that a random set of points is an $m$-blocking set, and the event that there is an incidence between a random set of points and a random set of $m$-flats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021).