Maximal arithmetic progressions in random subsets

Itai Benjamini, Ariel Yadin, Ofer Zeitouni

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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 logN/ log2 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)/ logN converges almost surely to 2/log 2, while W(N)/ logN does not converge almost surely (and in particular, limsup W(N)/ logN ≥ 3/ log 2).

Original languageEnglish
Pages (from-to)365-376
Number of pages12
JournalElectronic Communications in Probability
StatePublished - 1 Jan 2007
Externally publishedYes


  • Arithmetic progression
  • Chen-Stein method
  • Dependency graph
  • Extreme type limit distribution
  • Random subset

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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