Deviation probabilities for arithmetic progressions and other regular discrete structures

TL;DR

This paper establishes bounds on the probability that the count of edges in regular hypergraphs induced by random subsets deviates significantly from its expected value, with applications to arithmetic progressions and random vertex subsets.

Contribution

It introduces new probabilistic bounds for deviations in regular hypergraphs, extending to structures like arithmetic progressions and random vertex selections.

Findings

Derived bounds for deviation probabilities in regular hypergraphs.

Applied results to arithmetic progressions in cyclic groups.

Extended findings to random vertex subset models.

Abstract

Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $\mathcal{H}$ satisfies some regularity condition we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the cyclic group $\mathbb{Z}_N$. Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$.