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ON BADLY APPROXIMABLE COMPLEX NUMBERS

Published online by Cambridge University Press:  29 March 2010

R. ESDAHL-SCHOU  and
S. KRISTENSEN
R. ESDAHL-SCHOU
Affiliation:
Department of Mathematical Sciences, Faculty of Science, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark e-mail: estel@imf.au.dk
S. KRISTENSEN
Affiliation:
Department of Mathematical Sciences, Faculty of Science, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark e-mail: sik@imf.au.dk
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Abstract

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We show that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in $\(\mathbb{Q}(\sqrt{-D})\)$, where D ∈ {1, 2, 3, 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.

Keywords

11J83

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 349 - 355
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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