## Electronic Journal of Probability

### Choices, intervals and equidistribution

Matthew Junge

#### Abstract

We give a sufficient condition for a random sequence in [0,1] generated by a Psi-process to be equidistributed. The condition is met by the canonical example - the max-2 process - where the $n$th term is whichever of two uniformly placed points falls in the larger gap formed by the previous $n$-1 points. Also, we deduce equidistribution for an interpolation of the min-2 and max-2 processes that is biased towards min-2, as well as more general interpolations.  This solves an open problem from Itai Benjamini, Pascal Maillard and Elliot Paquette.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 97, 18 pp.

Dates
Accepted: 16 September 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067203

Digital Object Identifier
doi:10.1214/EJP.v20-4191

Mathematical Reviews number (MathSciNet)
MR3399833

Zentralblatt MATH identifier
1338.60132

Subjects
Primary: Probability

Rights

#### Citation

Junge, Matthew. Choices, intervals and equidistribution. Electron. J. Probab. 20 (2015), paper no. 97, 18 pp. doi:10.1214/EJP.v20-4191. https://projecteuclid.org/euclid.ejp/1465067203

#### References

• Azar, Yossi; Broder, Andrei Z.; Karlin, Anna R.; Upfal, Eli. Balanced allocations. SIAM J. Comput. 29 (1999), no. 1, 180–200.
• Blum, J. R.; Mizel, V. J. A generalized Weyl equidistribution theorem for operators, with applications. Trans. Amer. Math. Soc. 165 (1972), 291–307.
• Jacek Cichoad, Marek Klonowski, Łukasz Krzywiecki, Bartłomiej Róźański, and Paweł‚ Zieliński, Random subsets of the interval and p2p protocols, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (Moses Charikar, Klaus Jansen, Omer Reingold, and José D.P. Rolim, eds.), Lecture Notes in Computer Science, vol. 4627, Springer Berlin Heidelberg, 2007, pp. 409–421 (English).
• Darling, D. A. On a class of problems related to the random division of an interval. Ann. Math. Statistics 24, (1953). 239–253.
• Ford, Kevin; Soundararajan, K.; Zaharescu, Alexandru. On the distribution of imaginary parts of zeros of the Riemann zeta function. II. Math. Ann. 343 (2009), no. 3, 487–505.
• Kakutani, Shizuo. A problem of equidistribution on the unit interval $[0,1]$. Measure theory (Proc. Conf., Oberwolfach, 1975), pp. 369–375. Lecture Notes in Math., Vol. 541, Springer, Berlin, 1976.
• L. Kuipers and H. Niederreiter, phUniform distribution of sequences, Dover Books on Mathematics, Dover Publications, 2006.
• Luczak, Malwina J.; McDiarmid, Colin. On the power of two choices: balls and bins in continuous time. Ann. Appl. Probab. 15 (2005), no. 3, 1733–1764.
• Lootgieter, J.-C. Sur la repartition des suites de Kakutani. II. (French) Ann. Inst. H. Poincare Sect. B (N.S.) 14 (1978), no. 3, 279–302.
• P. Maillard and E. Paquette, Choices and intervals, ArXiv e-prints (2014).
• Mitzenmacher, Michael; Richa, Andra W.; Sitaraman, Ramesh. The power of two random choices: a survey of techniques and results. Handbook of randomized computing, Vol. I, II, 255–312, Comb. Optim., 9, Kluwer Acad. Publ., Dordrecht, 2001.
• Pyke, Ronald. The asymptotic behavior of spacings under Kakutani's model for interval subdivision. Ann. Probab. 8 (1980), no. 1, 157–163.
• Rudin, Walter. Principles of mathematical analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Dusseldorf, 1976. x+342 pp.
• Vaughan, R. C. On the distribution of $\alpha p$ modulo $1$. Mathematika 24 (1977), no. 2, 135–141.
• Hermann Weyl, Über die gibbs'sche erscheinung und verwandte konvergenzphànomene, Rendiconti del Circolo Matematico di Palermo 30 (1910), no. 1, 377–407 (Italian).
• van Zwet, W. R. A proof of Kakutani's conjecture on random subdivision of longest intervals. Ann. Probability 6 (1978), no. 1, 133–137.