Sets of Exact Approximation Order by Complex rational numbers

For a nonincreasing function $psi$, let $textrm{Exact}(psi)$ 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 $textrm{Exact}(psi)$ when $psi(x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-tau/(1-2tau)$ when $tau=limsup_{xtoinfty}psi(x)x^2$ small enough.