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On badly approximable complex numbers

By R. Esdahl-Schou and S. Kristensen
No. 03, March 2009
We show that the set of complex numbers which are badly approximable by ratios of elements of $\mathbb{Z}[\sqrt{-D}]$, where $D \in \{1,2,3,5,7,11,19,43,67,163\}$ has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.
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