Sets of Exact Approximation Order by Complex rational numbers

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 $2-\tau \mathrm{/}(1-2\tau )$ when $\tau ={\mathrm{lim\hspace{0.17em}sup}}_{x\to \mathrm{\infty }}\psi (x)x^2$ small enough.