The algebraic groups leading to the Roth inequalities
type: Item
instance of: scholarly article
title: The algebraic groups leading to the Roth inequalities
zbMATH Open document ID: 1276.11120
author: Masami Fujimori
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2012-10-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: A general ROTH system \(S=(\ell_1,\dots,\ell_n,c(1),\dots,c(n))\) is given by \(n\) linearly independent linear forms \((\ell_1,\dots,\ell_n)\) in \(n\) variables with real algebraic coefficients and \(n\) real numbers \(c(1),\dots,c(n)\) such that \(c(1)+\dots+c(n)=0\) and such that, for each \(\delta>0\), there are only finitely many integral solutions to the system of inequalities \[ |\ell_q|<Q^{-c(q)-\delta}, \quad Q>1, \; 1\leq q\leq n. \] The special case \(n=2\), \(\ell_1(x_1,x_2)=x_1\), \(\ell_2(x_1,x_2)=\alpha x_1-x_2\) where \(\alpha\) is an irrational real algebraic number and \(c(1)=-1\), \(c(2)=1\) is called a classical ROTH system. To a general ROTH system \(S\) is attached a filtration \(F_S^\cdot V\) over \(\overline{\mathbb Q}\) of the \(\mathbb Q\)-vector space \(V=\mathbb Q x_1+\dots + \mathbb Q x_n\), defined as \[ F_S^{i}V=\sum_{c(q)\geq i}\overline{\mathbb Q} \ell_q \quad (i\in\mathbb R). \] In the case of a classical ROTH system attached to an irrational algebraic number \(\alpha\), this filtered space is denoted by \((\check{V}, F_\alpha^\cdot \check{V})\). The slope of a filtered vector space \((V, F^\cdot V)\) is \[ \mu(V)=\frac{1}{\dim_\mathbb Q V} \sum_{w\in \mathbb R} w \dim_{\overline{\mathbb Q}} {\mathrm{gr}}^w(F^{\cdot} V) \] where \( {\mathrm{gr}}^w(F^{\cdot} V)=F^wV/F^{w+}V\), \(F^{w+}V=\bigcup _{j>w}F^jV\). The filtration is semi stable if \(\mu(W)=\mu(V)\) for any nonzero \(\mathbb Q\)-vector subspace \(W\) of \(V\). The category of finite dimensional vector spaces over \(\mathbb Q\) with semistable filtration of slope zero is denoted by \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\). See Chap.~VI, Theorem 2B of [\textit{W. M. Schmidt}, Diophantine approximation. Berlin etc.: Springer-Verlag (1980; Zbl 0421.10019)]. See also [\textit{G. Faltings}, Proc. ICM '94. Vol. I. Basel: Birkhäuser, 648--655 (1995; Zbl 0871.14010)] and [\textit{B. Totaro}, Duke Math. J. 83, No. 1, 79--104 (1996; Zbl 0873.14019)]. Here is the main resu…
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 6099144
MaRDI profile type: Publication
OpenAlex ID: W2009904948
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DOI: 10.5802/JTNB.796
description: scientific article; zbMATH DE number 6099144