Sets of exact approximation order by complex rational numbers

Abstract

Given a non-increasing function $\psi$, let ${{\,\mathrm{Exact}\,}}(\psi )$ be the set of complex numbers which are approximable by complex rational numbers to order $\psi$ but to no better order. In this paper, we obtain the Hausdorff dimension and the packing dimension of ${{\,\mathrm{Exact}\,}}(\psi )$ when $\psi (x)=o(x^{-2})$. Moreover, without the condition $\psi (x)=o(x^{-2})$, we also prove that the Hausdorff dimension of ${{\,\mathrm{Exact}\,}}(\psi )$ is greater than $2-\tau /(1-2\tau )$ when $0<\tau =\limsup _{x\rightarrow +\infty }x^2\psi (x)$ is small enough.