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

作者：He, Yubin; Xiong, Ying^*

来源：Fractals, 2021, 29(07): 2150179.

DOI：10.1142/S0218348X21501796

• Hurwitz Continued Fraction
• Partial Quotients
• psi-Approximation
• Jarnik-Type Theorem
• Hausdorff Dimension
• Packing Dimension

For regular continued fraction, if a real number x and its rational approximation p/q satisfying vertical bar x - p/q vertical bar < 1/q(2), 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(psi) of complex numbers which are well approximated with the given bound psi 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 psi and its Hausdorff dimension is equal to that of the psi-approximable set W(psi) of complex numbers. As a consequence, we also obtain an analogue of the classical Jarnik theorem in real case.

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