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arXiv · 1805.03778

Threshold functions for substructures in random subsets of finite vector spaces

Changhao Chen·Catherine Greenhill
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Abstract

The study of substructures in random objects has a long history, beginning with Erd\H{o}s and R\'enyi'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.

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Changhao Chen, Catherine Greenhill. 2018-05-10. Threshold functions for substructures in random subsets of finite vector spaces. https://arxiv.org/abs/1805.03778

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