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On the nonarchimedean quadratic Lagrange spectra

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Abstract

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.

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Notes

  1. See Sect. 4 for a definition.

  2. See [16] for a comparison between the naive height of the algebraic number \(\alpha \) and the above complexity of \(\alpha \) in the Archimedean case.

  3. Actually, Theorem 1.6 of [9] is stated only for \(K= {\mathbb F}_q(Y)\), \(v=v_\infty \) and \(\Gamma =\Gamma _v\), but it has an analogous version for general \((K,v,\Gamma )\) by using [9, Proposition 1.5].

  4. With an arbitrary ordering and \(N=(q^{k+1}-q)^m\).

  5. With an arbitrary order and \(N'=q^{2k+1}-q^k\).

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Acknowledgements

This work was supported by the French-Finnish CNRS grant PICS No 6950. We thank a lot Yann Bugeaud for his comments on the first version of this paper, which allowed us to remove the unnecessary assumption of odd characteristic, and gave a negative solution to a conjecture we proposed on a general formula for the Hurwitz constants.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Jyväskylä, Finland

    Jouni Parkkonen

  2. Laboratoire de mathématique d’Orsay, UMR 8628 Univ. Paris-Sud et CNRS, Université Paris-Saclay, 91405, ORSAY, Cedex, France

    Frédéric Paulin

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  1. Jouni Parkkonen
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  2. Frédéric Paulin
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Correspondence to Frédéric Paulin.

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Parkkonen, J., Paulin, F. On the nonarchimedean quadratic Lagrange spectra. Math. Z. 294, 1065–1084 (2020). https://doi.org/10.1007/s00209-019-02300-1

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