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Local positivity and effective Diophantine approximation

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Abstract

In this paper we present a new approach to prove effective results in Diophantine approximation. This approach involves measures of local positivity of divisors combined with Faltings’s version of Siegel’s lemma instead of a zero estimate such as Dyson’s lemma. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with complex coefficients.

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Acknowledgements

I would like to thank Matteo Costantini, Jan-Hendrik Evertse, Nathan Grieve, Walter Gubler, Ariyan Javanpeykar, Lars Kühne, Victor Lozovanu, Martin Lüdtke, Marco Maculan, David McKinnon, Mike Roth, Paul Vojta and Jürgen Wolfart for many helpful discussions and suggestions. I am grateful to János Kollár and Jakob Stix for their comments on an earlier version of this paper. Furthermore, I would like to express my gratitude towards my former thesis advisor Alex Küronya for his support and many useful comments. Finally, I thank the anonymous referee for their helpful comments that improved the quality of the paper. The author was partially supported by the LOEWE grant “Uniformized Structures in Algebra and Geometry”.

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  1. Institut für Mathematik, Johannes Gutenberg-Universität, Staudingerweg 9, 55128, Mainz, Germany

    Matthias Nickel

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Nickel, M. Local positivity and effective Diophantine approximation. Abh. Math. Semin. Univ. Hambg. 92, 125–138 (2022). https://doi.org/10.1007/s12188-022-00260-8

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