Sets of Exact Approximation Order by Complex rational numbers

Abstract

For a nonincreasing function $\psi$, let $\text{Exact}(\psi )$ be the set of complex numbers that are approximable by complex rational numbers to order $\psi$ but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of $\text{Exact}(\psi )$ when $\psi (x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-\tau /(1-2\tau )$ when $\tau ={\mathrm{limsup}}_{x\to \infty }\psi (x)x^2$ small enough.

References in corpus (1)

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