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Maximal Arithmetic Progressions in Random Subsets


 
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1. Title Title of document Maximal Arithmetic Progressions in Random Subsets
 
2. Creator Author's name, affiliation, country Itai Benjamini; Weizmann Institute of Science
 
2. Creator Author's name, affiliation, country Ariel Yadin; Weizmann Institute of Science
 
2. Creator Author's name, affiliation, country Ofer Zeitouni; University of Minnesota
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) arithmetic progression; random subset; Chen-Stein method; dependency graph; extreme type limit distribution
 
3. Subject Subject classification 60C05
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2007-10-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1321
 
10. Identifier Digital Object Identifier 10.1214/ECP.v12-1321
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 12
 
12. Language English=en
 
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