The difference between the Hurwitz continued fraction expansions of a complex number and its rational approximations

TL;DR

This paper investigates the differences in Hurwitz continued fraction expansions between complex numbers and their rational approximations, revealing dimension results and an analogue of Jarník's theorem.

Contribution

It demonstrates that, unlike real continued fractions, the Hurwitz expansions of complex numbers and their rational approximations can differ significantly, with precise dimension characterizations.

Findings

Packing dimension of the set is always full.

Hausdorff dimension equals that of the $ ext{ψ}$-approximable set.

Established an analogue of Jarník's theorem for complex numbers.

Abstract

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, 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 Jarn\'ik Theorem in real case.