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Digit expansions of numbers in different bases.

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Title: Digit expansions of numbers in different bases.
Authors: Burrell, Stuart A.1 (AUTHOR) sb235@st-andrews.ac.uk, Yu, Han1,2 (AUTHOR) hy351@maths.cam.ac.uk
Source: Journal of Number Theory. Sep2021, Vol. 226, p284-306. 23p.
Subjects: Fractal dimensions, Number theory, Integers, Real numbers, Logical prediction
Abstract: A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 150525873
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PubTypeId: academicJournal
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  Data: Digit expansions of numbers in different bases.
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Sep2021, Vol. 226, p284-306. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Fractal+dimensions%22">Fractal dimensions</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink><br /><searchLink fieldCode="DE" term="%22Real+numbers%22">Real numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Logical+prediction%22">Logical prediction</searchLink>
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  Data: A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1016/j.jnt.2021.01.003
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 23
        StartPage: 284
    Subjects:
      – SubjectFull: Fractal dimensions
        Type: general
      – SubjectFull: Number theory
        Type: general
      – SubjectFull: Integers
        Type: general
      – SubjectFull: Real numbers
        Type: general
      – SubjectFull: Logical prediction
        Type: general
    Titles:
      – TitleFull: Digit expansions of numbers in different bases.
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            NameFull: Burrell, Stuart A.
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            NameFull: Yu, Han
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          Dates:
            – D: 01
              M: 09
              Text: Sep2021
              Type: published
              Y: 2021
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              Value: 226
          Titles:
            – TitleFull: Journal of Number Theory
              Type: main
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