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Maximal Arithmetic Progressions in Random Subsets

   
@article{ECP1321,
	author = {Itai Benjamini and Ariel Yadin and Ofer Zeitouni},
	title = {Maximal Arithmetic Progressions in Random Subsets},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {12},
	year = {2007},
	keywords = {arithmetic progression; random subset; Chen-Stein method; dependency graph; extreme type limit distribution},
	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.
},
	pages = {no. 35, 365-376},
	issn = {1083-589X},
	doi = {10.1214/ECP.v12-1321},    
        url = {http://ecp.ejpecp.org/article/view/1321}}