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K. Mahler’s conjecture: There exists no <italic>ξ</italic> ∈ ℝ+ such that the fractional parts {<italic>ξ</italic>(3/2)<italic>n</italic>} satisfy 0 <italic>≤</italic> {<italic>ξ</italic>(3/2)<italic>n</italic>} &lt; 1/2 for all <italic>n</italic> = 0, 1, 2,... Such a <italic>ξ</italic>, if exists, is called a Mahler’s <italic>Z</italic>-number. In this paper we prove that if <italic>ξ</italic> is a <italic>Z</italic>-number, then the sequence <italic>xn</italic> = {<italic>ξ</italic>(3/2)<italic>n</italic>}, <italic>n</italic> =1, 2,... has asymptotic distribution function <italic>c</italic>0(<italic>x</italic>), where <italic>c</italic>0(<italic>x</italic>)=1 for <italic>x</italic> ∈ (0, 1].
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K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 ≤ {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2)n}, n =1, 2,... has asymptotic distribution function c0(x), where c0(x)=1 for x ∈ (0, 1].

Mahler’s Conjecture on <italic>ξ</italic>(3/2)<italic>n</italic>mod 1

Uniform distribution theory

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

K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 ≤ {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if...

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Published Online: Feb 02, 2022

Page range: 49 - 70

Received: Feb 27, 2019

Accepted: Aug 29, 2021

DOI: https://doi.org/10.2478/udt-2021-0007

Keywords
distribution function, fractional part, -number

© 2021 Oto Strauch, published by Sciendo

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

Mahler’s Conjecture on ξ(3/2)nmod 1

Published Online: Feb 02, 2022

Page range: 49 - 70

Received: Feb 27, 2019

Accepted: Aug 29, 2021

DOI: https://doi.org/10.2478/udt-2021-0007

Keywordsdistribution function, fractional part, -number

© 2021 Oto Strauch, published by Sciendo

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

Mahler’s Conjecture on ξ(3/2)ⁿmod 1

Keywords
distribution function, fractional part, -number