IDEAS home Printed from https://ideas.repec.org/a/wsi/fracta/v29y2021i07ns0218348x21501796.html
   My bibliography  Save this article

The Difference Between The Hurwitz Continued Fraction Expansions Of A Complex Number And Its Rational Approximations

Author

Listed:
  • YUBIN HE

    (Department of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China)

  • YING XIONG

    (Department of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China)

Abstract

For regular continued fraction, if a real number x and its rational approximation p/q satisfying |x − p/q| < 1/q2, then, after deleting the last integer of the partial quotients of p/q, the sequence of the remaining partial quotients is a prefix of that of x. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set E(ψ) of complex numbers which are well approximated with the given bound ψ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound ψ and its Hausdorff dimension is equal to that of the ψ-approximable set W(ψ) of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

Suggested Citation

  • Yubin He & Ying Xiong, 2021. "The Difference Between The Hurwitz Continued Fraction Expansions Of A Complex Number And Its Rational Approximations," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(07), pages 1-16, November.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:07:n:s0218348x21501796
    DOI: 10.1142/S0218348X21501796
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0218348X21501796
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0218348X21501796?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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:wsi:fracta:v:29:y:2021:i:07:n:s0218348x21501796. 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: Tai Tone Lim (email available below). General contact details of provider: https://www.worldscientific.com/worldscinet/fractals .

    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.