On the nonarchimedean quadratic Lagrange spectra

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. 1

Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

“…be the union of the orbits of α and α σ under the projective action of PGL 2 (R). Given f in K \ (K ∪ Θ α ), Parkkonen and Paulin [20] introduced the quadratic approximation constant of f , defined by…”

“…In the present paper, we reprove many of their results by means of the theory of continued fractions in power series fields and, in addition, we establish several new results. We stress that, unlike in [20], we do not assume that the prime power q is odd (however, Frédéric Paulin informed me that their approach also works well in characteristic 2). In particular, we give alternative proofs of the following two theorems highlighted in [20].…”

“…We stress that, unlike in [20], we do not assume that the prime power q is odd (however, Frédéric Paulin informed me that their approach also works well in characteristic 2). In particular, we give alternative proofs of the following two theorems highlighted in [20]. Theorem 1.1 (Parkkonen and Paulin [20]).…”

“…In particular, we give alternative proofs of the following two theorems highlighted in [20]. Theorem 1.1 (Parkkonen and Paulin [20]). Let α be a quadratic power series in K.…”

“…(2) (Hall's ray) There exists an integer m α such that, for every integer n with n > m α , the real number q −n belongs to Sp(α). [20]). The Hurwitz constant of any quadratic power series in K, whose continued fraction expansion is eventually periodic with a period of length at most q − 1, is equal to q −2 .…”