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2007 Maximal Arithmetic Progressions in Random Subsets
Itai Benjamini, Ariel Yadin, Ofer Zeitouni
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Electron. Commun. Probab. 12: 365-376 (2007). DOI: 10.1214/ECP.v12-1321

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.

Citation

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Itai Benjamini. Ariel Yadin. Ofer Zeitouni. "Maximal Arithmetic Progressions in Random Subsets." Electron. Commun. Probab. 12 365 - 376, 2007. https://doi.org/10.1214/ECP.v12-1321

Information

Accepted: 14 October 2007; Published: 2007
First available in Project Euclid: 6 June 2016

zbMATH: 1133.60007
MathSciNet: MR2350574
Digital Object Identifier: 10.1214/ECP.v12-1321

Subjects:
Primary: 60C05

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