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Classification de Mahler et distances locales

Published online by Cambridge University Press:  17 April 2009

Patrice Philippon
Patrice Philippon
Affiliation:
Probèmes Diophantiens UniversitéP. et M. Curie T.45–46, 5 ème ét. 75252 Paris, Cedex 05 FranceFrance
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We consider the classifying set, denoted ℂ/˜ below, introduced by Mahler and we show that it can be endowed with non-discrete, Hausdorff topologies (even local distances) based on diophantine approximation properties of (complex) numbers. We then establish several results on the pointed topological space obtained, which could be dubbed Mahler's space (of first degree). We recover an analogue of Mahler's original classification by considering local distances to the special point (that is the class of all algebraic numbers). The main ingredients used here are diophantine properties in dimension zero over ℚ and non-standard analysis.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 2 , April 1994 , pp. 219 - 238
Copyright
Copyright © Australian Mathematical Society 1994

References

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