Sets of exact approximation order by complex rational numbers

Authors：He, YuBin; Xiong, Ying^*

Source：Mathematische Zeitschrift, 2022, 301(1): 199-223.

DOI：10.1007/s00209-021-02906-4

• Diophantine approximation
• Exact approximation
• Hausdorff dimension
• Packing dimension
• Hurwitz continued fraction

Given a non-increasing function psi, let 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 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 Exact(psi) is greater than 2 - tau/(1 - 2 tau) when 0 < tau = lim sup(x ->+infinity) x(2) psi(x) is small enough.

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