Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

Yann Bugeaud Fundamenta Mathematicae MSC: 11J06, 11J61, 11J70, 11R11. DOI: 10.4064/fm622-2-2019 Opublikowany online: 5 July 2019

Streszczenie

Let $\mathbb F_q$ be a finite field of order a positive power $q$ of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given quadratic power series in $\mathbb F_q((Y^{-1}))$, for the action by homographies and anti-homographies of ${\rm PGL}_2(\mathbb F_q[Y])$ on $\mathbb F_q((Y^{-1})) \cup \{\infty \}$. While Parkkonen and Paulin’s approach used geometric methods of group actions on Bruhat–Tits trees, ours is based on the theory of continued fractions in power series fields.

Autorzy

  • Yann BugeaudInstitut de Recherche Mathématique Avancée, U.M.R. 7501
    Université de Strasbourg et C.N.R.S.
    7, rue René Descartes
    67084 Strasbourg, France
    e-mail

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