Modular group, continued fractions and Diophantine approximation in characteristic \(p\)
type: Item
instance of: scholarly article
title: Modular group, continued fractions and Diophantine approximation in characteristic \(p\)
zbMATH Open document ID: 1033.20026
DOI: 10.1023/A:1021270631563
author: Frédéric Paulin
published in: Geometriae Dedicata
publication date: 2003-03-11T00:00:00Z
review text: Continued fraction expansions of formal Laurent series in \(x^{-1}\) over a finite field were already used by E. Artin in his Ph.D. dissertation, published in 1924; the role of the ``integer'' part here is played by the ``polynomial'' part. The Diophantine and metric properties of these continued fractions have recently been studied, see say \textit{W. M. Schmidt} [Acta Arith. 95, No. 2, 139-166 (2000; Zbl 0987.11041)], and \textit{V. Berthé} and \textit{H. Nakada} [Expo. Math. 18, No. 4, 257-284 (2000; Zbl 1024.11050)]. Here, the author gives a clear geometric interpretation of this continued fraction expansion in terms of geodesics of the Bruhat-Tits tree for the special linear group of the Laurent series field. Results include geometric proofs of (quantitative versions of) the facts: convergents give best approximations; quadratics have periodic expansions; and, elements algebraic over the Laurent series field are poorly approximated.
reviewed by: Thomas A. Schmidt
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 11J61
zbMATH DE Number: 1879686
zbMATH Keywords: Bruhat-Tits trees
zbMATH Keywords: formal Laurent series fields
zbMATH Keywords: continued fraction expansions
zbMATH Keywords: special linear groups
zbMATH Keywords: Diophantine approximation
MaRDI profile type: Publication
full work available at URL: a:1021270631563
OpenAlex ID: W146286979
description: scientific article; zbMATH DE number 1879686