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Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties.

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Title: Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties.
Authors: Tian, Zhigang1 (AUTHOR) 202010106099@mail.scut.edu.cn, Fang, Lulu1,2 (AUTHOR) fanglulu@njust.edu.cn
Source: Journal of Number Theory. Mar2026, Vol. 280, p947-972. 26p.
Subjects: Continued fractions, Approximation theory, Mathematical expansion, Mathematics theorems, Baire spaces, Lebesgue measure, Irrational numbers
Abstract: In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal 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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  Data: <searchLink fieldCode="AR" term="%22Tian%2C+Zhigang%22">Tian, Zhigang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> 202010106099@mail.scut.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Fang%2C+Lulu%22">Fang, Lulu</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> fanglulu@njust.edu.cn</i>
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  Data: <searchLink fieldCode="DE" term="%22Continued+fractions%22">Continued fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+theory%22">Approximation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+expansion%22">Mathematical expansion</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+theorems%22">Mathematics theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Baire+spaces%22">Baire spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Lebesgue+measure%22">Lebesgue measure</searchLink><br /><searchLink fieldCode="DE" term="%22Irrational+numbers%22">Irrational numbers</searchLink>
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  Data: In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal dimension. [ABSTRACT FROM AUTHOR]
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  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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      – Type: doi
        Value: 10.1016/j.jnt.2025.09.006
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 26
        StartPage: 947
    Subjects:
      – SubjectFull: Continued fractions
        Type: general
      – SubjectFull: Approximation theory
        Type: general
      – SubjectFull: Mathematical expansion
        Type: general
      – SubjectFull: Mathematics theorems
        Type: general
      – SubjectFull: Baire spaces
        Type: general
      – SubjectFull: Lebesgue measure
        Type: general
      – SubjectFull: Irrational numbers
        Type: general
    Titles:
      – TitleFull: Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties.
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            NameFull: Tian, Zhigang
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            NameFull: Fang, Lulu
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          Dates:
            – D: 01
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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              Value: 280
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            – TitleFull: Journal of Number Theory
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