Maximal Arithmetic Progressions in Random Subsets

Abstract

Let $U(N)$ denote the maximal length of arithmetic progressions in a random uniform subset of $\{0,1\}^N$. By an application of the Chen-Stein method, we show that $U(N)- 2 \log(N)/\log(2)$ converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length $W(N)$ of arithmetic progressions (mod $N$). When considered in the natural way on a common probability space, we observe that $U(N)/\log(N)$ converges almost surely to $2/\log(2)$, while $W(N)/\log(N)$ does not converge almost surely (and in particular, $\limsup W(N)/\log(N)$ is at least $3/\log(2)$).

An Erratum is available in ECP volume 17 paper number 18.