THE DIFFERENCE BETWEEN THE HURWITZ CONTINUED FRACTION EXPANSIONS OF A COMPLEX NUMBER AND ITS RATIONAL APPROXIMATIONS

Yubin He () and Ying Xiong
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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

FRACTALS (fractals), 2021, vol. 29, issue 07, 1-16

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.

Keywords: Hurwitz Continued Fraction; Partial Quotients; ψ-Approximation; Jarník-Type Theorem; Hausdorff Dimension; Packing Dimension (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501796

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