## Electronic Journal of Probability

### On the maximal length of arithmetic progressions

#### Abstract

This paper is a continuation of a paper by Benjamini, Yadin and Zeitouni's on maximal arithmetic progressions in random subsets. In this paper the asymptotic distributions of the maximal arithmetic progressions and arithmetic progressions modulo $n$ relative to an independent Bernoulli sequence with parameter $p$ are given. The errors are estimated by using the Chen-Stein method. Then the almost sure limit behaviour of these statistics is discussed. Our work extends the previous results and gives an affirmative answer to the conjecture raised at the end of that paper.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 79, 21 pp.

Dates
Accepted: 31 August 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064304

Digital Object Identifier
doi:10.1214/EJP.v18-2018

Mathematical Reviews number (MathSciNet)
MR3101645

Zentralblatt MATH identifier
1285.60022

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Rights

#### Citation

Zhao, Minzhi; Zhang, Huizeng. On the maximal length of arithmetic progressions. Electron. J. Probab. 18 (2013), paper no. 79, 21 pp. doi:10.1214/EJP.v18-2018. https://projecteuclid.org/euclid.ejp/1465064304

#### References

• Alon, Noga; Spencer, Joel H. The probabilistic method. Third edition. With an appendix on the life and work of Paul ErdÅ‘s. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2008. xviii+352 pp. ISBN: 978-0-470-17020-5
• Arratia, R.; Goldstein, L.; Gordon, L. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17 (1989), no. 1, 9–25.
• An introduction to Stein's method. Lectures from the Meeting on Stein's Method and Applications: a Program in Honor of Charles Stein held at the National University of Singapore, Singapore, July 28â€“August 31, 2003. Edited by A. D. Barbour and Louis H. Y. Chen. Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, 4. Singapore University Press, Singapore; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xii+225 pp. ISBN: 981-256-330-X
• Benjamini, Itai; Yadin, Ariel; Zeitouni, Ofer. Maximal arithmetic progressions in random subsets. Electron. Comm. Probab. 12 (2007), 365–376.
• Erdős, Paul; Rényi, Alfréd. On a new law of large numbers. J. Analyse Math. 23 1970 103–111.
• Erdős, P.; Révész, P. On the length of the longest head-run. Topics in information theory (Second Colloq., Keszthely, 1975), pp. 219–228. Colloq. Math. Soc. Janos Bolyai, Vol. 16, North-Holland, Amsterdam, 1977.
• Erdős, Paul; Turán, Paul. On Some Sequences of Integers. J. London Math. Soc. S1-11 no. 4, 261.
• Gordon, Louis; Schilling, Mark F.; Waterman, Michael S. An extreme value theory for long head runs. Probab. Theory Relat. Fields 72 (1986), no. 2, 279–287.
• Túri, József. Limit theorems for the longest run. Ann. Math. Inform. 36 (2009), 133–141.
• Kohayakawa, Yoshiharu; Łuczak, Tomasz; Rödl, Vojtŭch. Arithmetic progressions of length three in subsets of a random set. Acta Arith. 75 (1996), no. 2, 133–163.
• Móri, Tamás F. The a.s. limit distribution of the longest head run. Canad. J. Math. 45 (1993), no. 6, 1245–1262.
• Roth, K. F. On certain sets of integers. J. London Math. Soc. 28, (1953). 104–109.
• Szemerédi, E. On sets of integers containing no $k$ elements in arithmetic progression. Collection of articles in memory of Juriĭ­ Vladimirovič Linnik. Acta Arith. 27 (1975), 199–245.
• Szemerédi, E.; Vu, V. H. Finite and infinite arithmetic progressions in sumsets. Ann. of Math. (2) 163 (2006), no. 1, 1–35.
• Tao, Terence. What is good mathematics? Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 4, 623–634.