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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a theorem of Mahler
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by N. K. Meher, K. Senthil Kumar and R. Thangadurai PDF
Proc. Amer. Math. Soc. 145 (2017), 4607-4615 Request permission

Abstract:

Let $b\geq 2$ be an integer and $\alpha$ a non-zero real number written in base $b$. In 1973, Mahler proved the following result: Let $\alpha$ be an irrational number written in base $b$ and let $n\geq 1$ be a given integer. Let $B = b_0b_1\ldots b_{n-1}$ be a given block of digits in base $b$ of length $n$. Then, there exists an integer $X$ with $1\leq X < b^{2n+1}$ such that $B$ occurs infinitely often in the base $b$ representation of the fractional part of $X\alpha$. In this short note, we deal with some conditional quantitative version of this result.
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Additional Information
  • N. K. Meher
  • Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India
  • Email: nabinmeher@hri.res.in
  • K. Senthil Kumar
  • Affiliation: Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai, 603103, India
  • Address at time of publication: National Institute of Science Education and Research, HBNI, P.O. Jatni, Khurda 752050, Odisha, India
  • MR Author ID: 1050158
  • Email: senthil@niser.ac.in
  • R. Thangadurai
  • Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India
  • Email: thanga@hri.res.in
  • Received by editor(s): September 25, 2016
  • Received by editor(s) in revised form: November 1, 2016, and November 28, 2016
  • Published electronically: May 26, 2017

  • Dedicated: Dedicated to Michel Waldschmidt on his 70th birthday
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4607-4615
  • MSC (2010): Primary 11K16
  • DOI: https://doi.org/10.1090/proc/13616
  • MathSciNet review: 3691980