The Difference Between the Hurwitz Continued Fraction Expansions of a Complex Number and Its Rational Approximations

Abstract

For regular continued fraction, if a real number [Formula: see text] and its rational approximation [Formula: see text] satisfying [Formula: see text], then, after deleting the last integer of the partial quotients of [Formula: see text], the sequence of the remaining partial quotients is a prefix of that of [Formula: see text]. 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 [Formula: see text] of complex numbers which are well approximated with the given bound [Formula: see text] 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 [Formula: see text] and its Hausdorff dimension is equal to that of the [Formula: see text]-approximable set [Formula: see text] of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.