Threshold functions for substructures in random subsets of finite vector spaces

Abstract

The study of substructures in random objects has a long history, beginning with Erdős and Rényi'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.

Threshold functions for incidence properties in finite vector spaces

“…Theorem 1.1 (Chen-Greenhill, [4]). Let F be a family of ℓ-subsets of F n q , and X be a random subset chosen as above.…”

“…As consequences, they obtained threshold functions for several precise geometric structures; for examples, non-trivial k-term arithmetic progressions, non-trivial parallelograms, and non-trivial right angles. We refer the interested reader to [4] for more details.…”

“…Here, we study thresholds for properties of point sets in finite vector spaces. Hundreds of papers have been written on extremal questions in finite geometry, but we are only aware of a single work on thresholds in this context, by Chen and Greenhill [4]. Our contributions are to improve the main result of Chen and Greenhill, and to study thresholds for several properties defined by the incidences between points and affine subspaces.…”