Sets of Exact Approximation Order by Complex rational numbers

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 $\psi$, 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 $2-\tau/(1-2\tau)$ when $\tau=\limsup_{x\to\infty}\psi(x)x^2$ small enough.