Threshold functions for incidence properties in finite vector spaces

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 ℓ-rich m-flat with regard to a random set of points in F n q , 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 mflats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021).