Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

Abstract

Let 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 F q ((Y −1 )), for the action by homographies and anti-homographies of PGLWhile 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.

On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

“…In 2016, M. Artigiani, L. Marchese, LAGRANGE AND MARKOV DYNAMICAL SPECTRA 189 C. Ulcigrai [2] showed that Veech surfaces also have a Hall's ray. In the context of non-archimedean quadratic Lagrange spectra, some new results is this direction can be found in [32,5].…”

Markov and Lagrange spectra for Laurent series in $1/T$ with rational coefficients

“…In other words it measures the quality of the approximation of α in terms of the exponent. Recently, some work has been done on the Lagrange spectrum in the setting of formal Laurent series over finite fields, by Parkkonen and Paulin, and Bugeaud in [7] and [1], respectively. They define and study the nonarchimedian quadratic Lagrange spectrum, whose elements are approximations by the orbit of a given quadratic irrational in F q ((T −1 )).…”

On the nonarchimedean quadratic Lagrange spectra

“…After the first version of this paper was posted on ArXiv, Yann Bugeaud [Bug2] has given a completely different proof of the above results (except the generalisation to function fields), and proved several new theorems giving a more precise description of these spectra. In particular, he proved that all approximation constants for a given quadratic irrational are attained on the other quadratic irrationals, that for every m ≥ 2 there exists β ∈ K (2) such that max Sp(β) = q −m , and that for all ∈ N, there exists β ∈ K (2) such that Sp(β) contains exactly gaps.…”