IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i1p127-d716275.html
   My bibliography  Save this article

On the Eventually Periodic Continued β -Fractions and Their Lévy Constants

Author

Listed:
  • Qian Xiao

    (School of Mathematics, South China University of Technology, Guangzhou 510640, China)

  • Chao Ma

    (Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China)

  • Shuailing Wang

    (School of Mathematics, South China University of Technology, Guangzhou 510640, China)

Abstract

In this paper, we consider continued β -fractions with golden ratio base β . We show that if the continued β -fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z [ β ] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β -fraction expansion is dense in [ c , + ∞ ), where c = 1 2 log β + 2 − 5 β + 1 2 .

Suggested Citation

  • Qian Xiao & Chao Ma & Shuailing Wang, 2022. "On the Eventually Periodic Continued β -Fractions and Their Lévy Constants," Mathematics, MDPI, vol. 10(1), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:1:p:127-:d:716275
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/1/127/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/1/127/
    Download Restriction: no
    ---><---

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:1:p:127-:d:716275. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.