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

Threshold functions for substructures in random subsets of finite vector spaces