Diophantine approximation and nonstandard ultrametric distances
type: Item
instance of: scholarly article
title: Diophantine approximation and nonstandard ultrametric distances
zbMATH Open document ID: 1166.11024
author: Sandra Delaunay
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73661
review text: In his paper ``Classification de Mahler et distances locales'' [Bull. Aust. Math. Soc. 49, No. 2, 219--238 (1994; Zbl 0799.11019)], \textit{P. Philippon} considered the quotient space \(KM_1\) of \(\mathbb C\) by the equivalence relation \(\alpha\sim\beta\) if \(\alpha\) is algebraic over \({\mathbb Q}(\beta)\) and \(\beta\) is algebraic over \({\mathbb Q}(\alpha)\). This relation classifies complex numbers according to algebraic independence. Here, the author works out a similar theory for two other quotient spaces. The first one \(QH_1\) classifies real numbers according to rational dependence, the equivalence relation is \(\alpha\sim\beta\) if \(\beta=(a\alpha+b)/(c\alpha+d)\) with rational integers \(a,b,c,d\) satisfying \(ad-bc\not=0\). The second one \(QL_1\) classifies real numbers according to linear dependence, the equivalence relation is \(\alpha\sim\beta\) if \(q\beta=p\alpha\) with non-zero rational integers \(p,q\). She considers not only the classical case of real numbers, but also the field of formal Laurent series in one variable in any characteristic. She defines uniform topologies on these quotient sets which reflect Diophantine properties.
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 11U10
zbMATH DE Number: 5016732
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: classification of numbers
zbMATH Keywords: ultrametric non-standard distances
zbMATH Keywords: hyperreals
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
OpenAlex ID: W2330560499
cites work: Q5781743
cites work: Q3241280
cites work: Q3708853
cites work: Q4218534
cites work: Thue's Theorem in positive characteristic.
cites work: Nombres transcendants
cites work: Classification de Mahler et distances locales
cites work: Q4203379
DOI: 10.5802/AFST.1106
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description: scientific article; zbMATH DE number 5016732