Deviation probabilities for arithmetic progressions and other regular discrete structures

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 $p$.