Threshold functions for substructures in random subsets of finite vector spaces

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