A Note on the Distributions of (log n)mod 1

Arno Berger

doi:10.2478/udt-2022-0013

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A Note on the Distributions of (log n)mod 1

Uniform distribution theory

Volume 17 (2022): Issue 2 (December 2022)

By: Arno Berger

Open Access

|Dec 2022

Abstract

For sequences sufficiently close to (a log n), with an arbitrary real constant a, this note describes the precise asymptotics of the associated empirical distributions modulo one, with respect to the Kantorovich metric as well as a discrepancy-style metric. In particular, the note demonstrates how these asymptotics depend on a in a delicate, discontinuous way. The results strengthen and complement known facts in the literature.

References

[1] BECK, J.: Randomness of the square root of 2 and the giant leap, Part 1,Period. Math. Hungar. 60 (2010), 137–242.10.1007/s10998-010-2137-9
Search in Google Scholar Back to article
[2] BERGER, A.: Circling the uniform distribution (in preparation).
Search in Google Scholar Back to article
[3] BERGER, A.—HILL, T. P.: An Introduction to Benford’s Law. Princeton University Press, Princeton, NJ, 2015.
Search in Google Scholar Back to article
[4] BROWN, L.—STEINERBERGER, S.: On the Wasserstein distance between classical sequences and the Lebesgue measure, Trans. Amer. Math. Soc. 373 (2020), 8943–8962.10.1090/tran/8212
Search in Google Scholar Back to article
[5] BURTON, R.—DENKER, M.: On the central limit theorem for dynamical systems,Trans. Amer. Math. Soc. 302 (1987), 715–726.10.1090/S0002-9947-1987-0891642-6
Search in Google Scholar Back to article
[6] CABRELLI, C. A.—MOLTER, U. M.: The Kantorovich metric for probability measures on the circle, J. Comput. Appl. Math. 57 (1995), 345–361.10.1016/0377-0427(93)E0213-6
Search in Google Scholar Back to article
[7] CIGLER, J.: Asymptotische Verteilung reeller Zahlen mod 1, Monatsh. Math. 64 (1960), 201–225.10.1007/BF01361535
Search in Google Scholar Back to article
[8] DUDLEY, R. M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics Vol. 74, Cambridge University Press, Cambridge, 2002.
Search in Google Scholar Back to article
[9] FLEHINGER, B. J.: On the probability that a random integer has initial digit A,Amer. Math. Monthly 73 (1966), 1056–1061.10.1080/00029890.1966.11970894
Search in Google Scholar Back to article
[10] FUKUYAMA, K.: The central limit theorem for Riesz–Raikov sums, Probab. Theory Related Fields 100 (1994), 57–75.10.1007/BF01204953
Search in Google Scholar Back to article
[11] FUKUYAMA, K.: The central limit theorem for ∑ f (θⁿx)g(θⁿ^² x), Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1335–1353.
Search in Google Scholar Back to article
[12] GIULIANO ANTONINI, R.—STRAUCH, O.: On weighted distribution functions of sequences, Unif. Distrib. Theory 3 (2008), 1–18.
Search in Google Scholar Back to article
[13] GRAHAM, C.: Irregularity of distribution in Wasserstein distance, J. Fourier Anal. Appl. 26 (2020), no. 5, Paper no. 75, 21 pp.
Search in Google Scholar Back to article
[14] HUTSON, H. L.: On the distribution of log(p), Int. J. Pure Appl. Math. 7 (2003), 499–508.
Search in Google Scholar Back to article
[15] KEMPERMAN, J. H. B.: Distributions modulo 1 of slowly changing sequences,Nieuw Arch. Wisk. 21 (1973), 138–163.
Search in Google Scholar Back to article
[16] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. Wiley--Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
Search in Google Scholar Back to article
[17] NIEDERREITER, H.: Distribution mod 1 of monotone sequences, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), 315–327.10.1016/1385-7258(84)90031-3
Search in Google Scholar Back to article
[18] OHKUBO, Y.: On sequences involving primes, Unif. Distrib. Theory 6 (2011), 221–238.
Search in Google Scholar Back to article
[19] OHKUBO, Y.—STRAUCH, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory 11 (2016), 23–45.10.1515/udt-2016-0003
Search in Google Scholar Back to article
[20] OHKUBO, Y.—STRAUCH, O.: Distribution of leading digits of numbers II, Unif. Distrib. Theory 14 (2019), 19–42.10.2478/udt-2019-0003
Search in Google Scholar Back to article
[21] STEINERBERGER, S.: Wasserstein distance, Fourier series and applications, Monatsh. Math. 194 (2021), 305–338.10.1007/s00605-020-01497-2
Search in Google Scholar Back to article
[22] STRAUCH, O.— BLAŽEKOVÁ, O.: Distribution of the sequence p_n/n mod 1, Unif. Distrib. Theory 1 (2006), 45–63.
Search in Google Scholar Back to article
[23] WINKLER, R.: On the distribution behaviour of sequences,Math. Nachr. 186 (1997), 303–312.10.1002/mana.3211860118
Search in Google Scholar Back to article
[24] WINTNER, A.: On the cyclical distribution of the logarithms of the prime numbers, Q. J. Math., Oxf. Ser. 6 1 (1935), 65–68.10.1093/qmath/os-6.1.65
Search in Google Scholar Back to article
[25] XU, C.: The distributional asymptotics mod1 of (log_bn), Unif. Distrib. Theory 14 (2019), 105–122.10.2478/udt-2019-0007
Search in Google Scholar Back to article
[26] XU, C.—BERGER, A.: Best finite constrained approximations of one-dimensional probabilities, J. Approx. Theory 244 (2019), 1–36.10.1016/j.jat.2019.03.005
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/udt-2022-0013 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 77 - 100

Submitted on: Apr 29, 2022

Accepted on: Jun 21, 2022

Published on: Dec 12, 2022

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Slowly changing sequence,

distribution modulo one,

Kantorovich metric,

discrepancy

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2022 Arno Berger, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 17 (2022): Issue 2 (December 2022)