Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

2018/04/10 by Yann Bugeaud, Bugeaud, Yann
Mathematics · #11J06 #11J61 #11J70 #11R11 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories #math.NT #msc:11J06 #msc:11J61 #msc:11J70 #msc:11R11

paper · pdf · doi:10.48550/arxiv.1804.03566

18 pages

arxiv created 2018/04/10 · openalex publication_date 2018/04/10 · arxiv updated 2018/04/11 · openalex created_date 2023/02/16 · openalex updated_date 2026/07/28

Abstract

Let \bf 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 \bf F_q((Y^-1)), for the action by homographies and anti-homographies of \rm PGL₂(\bf F_q[Y]) on \bf F_q((Y^-1)) ∪ \∞\. While their approach used geometric methods of group actions on Bruhat--Tits trees, ours is based on the theory of continued fractions in power series fields.

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