Diophantine approximation and parametric geometry of numbers
type: Item
instance of: scholarly article
title: Diophantine approximation and parametric geometry of numbers
zbMATH Open document ID: 1264.11056
author: Leonhard Summerer
author: Wolfgang M. Schmidt
published in: Monatshefte für Mathematik
publication date: 2013-01-28T00:00:00Z
review text: Dirichlet's Theorem says that for \(n>1\), real numbers \(\xi_1,\dots,\xi_{n-1}\) and \(Q\geq 1\), there is a nonzero point \(\mathbf y=(y, y_1,\dots,y_{n-1})\in\mathbb Z^n\) having \(|y|\leq Q, |\xi_i y-y_i|\leq Q^{-1(n-1)}\) for \(i=1,\dots, n-1\). On the other hand let \(\Lambda\subset \mathbb R^n\) be a lattice and \(\mathcal K\) a closed symmetric convex body in \(\mathbb R^n\) with \(vol(\mathcal K)\geq 2^n \det \Lambda\). From Minkowski, \(\mathcal K\) contains a nonzero lattice point. Now, let \(\Lambda :=\Lambda(\xi)=\Lambda(\xi_1,\dots,\xi_{n-1})\) be the lattice of points \((y,\xi_1 y-y_1,\dots,\xi_{n-1} y-y_{n-1})\) where \(\mathbf y=(y,y_1,\dots,y_{n-1})\) runs through \(\mathbb Z^n\). Let \(\mathcal K(Q)\) consists of points \((x_1,x_2,\dots,x_n)\in\mathbb R^n\) with \(|x_1|\leq Q, |x_i|\leq Q^{-1/(n-1)}\) for \(i=2,\dots,n\). Then \(\det(\Lambda)=1\), \(vol(\mathcal K(Q))=2^n\) so that \(\mathcal K(Q)\) contains a nonzero point. Let us define \(q\) by \(Q=e^{(n-1)q}\). Then \(|x_1|\leq e^{(n-1)q}, |x_i|\leq e^{-q}\) for \(i=2,\dots,n\). Let us consider the convex body \(\mathcal K(q)=T^q(\mathcal K)\) parametrized by \(q\) where \(\mathcal K\) is a given symmetric convex body and \(T: \mathbb R^n\to \mathbb R^n\) is the map with \(T(x_1,x_2,\dots,x_n)=(e^{-(n-1)}x_1, e^{-1} x_2,\dots, e^{-1} x_n)\). Let \(\lambda_i(q)=\lambda_i(\Lambda,\mathcal K(q))=\lambda_i(\Lambda, \mathcal K,q)\) be the Minkowki successive minima of the convex body \(\mathcal K(q)\) seen as functions of \(q\), \(L_i=\log \lambda_i (q)\) and \(\phi_i(q)=L_i(q)/q, \;\overline{\phi_i}= \lim\sup _{q\to\infty} \phi_i(q)\) and \(\underline{\phi_i}= \lim\inf _{q\to\infty} \phi_i(q)\). A lattice \(\Lambda\) will be called \textit{proper} if for every \(j,\;1\leq j\leq n,\) there are arbitrarily large values of \(q\) with \(L_j(q)=L_{j+1}(q)\). Observe that, from [\textit{W. M. Schmidt} and \textit{L. Summerer}, Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)], \(\Lambda( \xi_1,\dot…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11J13
zbMATH DE Number: 6131821
zbMATH Keywords: successive minima
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: approximation constants
MaRDI profile type: Publication
full work available at URL: s00605-012-0391-z
OpenAlex ID: W2078998846
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DOI: 10.1007/S00605-012-0391-Z
description: scientific article; zbMATH DE number 6131821