<abstract xmlns="http://www.w3.org/1999/xhtml">This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (log<italic>b n</italic>) with <italic>b</italic> ∈ 𝕅 \ {1}. It is known that (logb<italic>n</italic>) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log <italic>b.</italic> An improved upper estimate (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt><mo>/</mo><mi>N</mi></mrow></math><tex-math>\sqrt {\log N} /N</tex-math></alternatives></inline-formula>) is obtained for the rate of convergence with respect to (w. r. t.)the <italic>Kantorovich metric</italic> on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log <italic>N/N</italic>)w. r. t. the <italic>Kantorovich metric</italic> on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the <italic>discrepancy metric</italic> (or the <italic>Kolmogorov metric</italic>) turns out to be (log <italic>N/N</italic>) as well, which verifies that an upper bound for this rate derived in [O<sc>hkubo</sc>, Y.—S<sc>trauch</sc>, O.: <italic>Distribution of leading digits of numbers</italic>, Unif. Distrib. Theory, <bold>11</bold> (2016), no.1, 23–45.] is sharp.</abstract>

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (logN/N\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

The Distributional Asymptotics Mod 1 of (log<italic>b n</italic>)

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The Distributional Asymptotics Mod 1 of (log)

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not...

The Distributional Asymptotics Mod 1 of (log_b n)

Published Online: Mar 27, 2020

Page range: 105 - 122

Received: Oct 02, 2017

Accepted: Sep 12, 2018

DOI: https://doi.org/10.2478/udt-2019-0007

Keywords
Uniformly distributed modulo one sequence, slowly changing sequence, rate of convergence, Kantorovich metric, discrepancy, probability measures

© 2019 Chuang Xu, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The Distributional Asymptotics Mod 1 of (logb n)

Published Online: Mar 27, 2020

Page range: 105 - 122

Received: Oct 02, 2017

Accepted: Sep 12, 2018

DOI: https://doi.org/10.2478/udt-2019-0007

KeywordsUniformly distributed modulo one sequence, slowly changing sequence, rate of convergence, Kantorovich metric, discrepancy, probability measures

© 2019 Chuang Xu, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The Distributional Asymptotics Mod 1 of (log_b n)

Keywords
Uniformly distributed modulo one sequence, slowly changing sequence, rate of convergence, Kantorovich metric, discrepancy, probability measures