Deviation probabilities for arithmetic progressions and irregular 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 the degrees of vertices in $\mathcal{H}$ vary significantly 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 $\{1,\dots, N\}$. Furthermore, our main theorem allows us to deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$. In this setting our result on arithmetic progressions extends a result of Bhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connections to related central limit theorems.