Sets of Exact Approximation Order by Complex rational numbers
Yubin He, Ying Xiong

TL;DR
This paper investigates the size and structure of sets of complex numbers that are approximable by rational numbers to a specific order, providing dimension results under certain decay conditions of the approximation function.
Contribution
It determines the Hausdorff and packing dimensions of these sets for functions decaying faster than x^{-2} and establishes a lower bound on the Hausdorff dimension based on the limsup of the approximation order.
Findings
Hausdorff and packing dimensions computed for ext{Exact}( ext{ extpsi}) when extpsi(x)=o(x^{-2})
Lower bound of Hausdorff dimension exceeds 2- au/(1-2 au) for small au
Dimension results depend on the decay rate of the approximation function extpsi
Abstract
For a nonincreasing function , let be the set of complex numbers that are approximable by complex rational numbers to order but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of when . We also prove that the lower bound of the Hausdorff dimension is greater than when small enough.
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