Small remainder of a vector to suitable modulus
type: Item
instance of: scholarly article
title: Small remainder of a vector to suitable modulus
zbMATH Open document ID: 0841.11036
DOI: 10.1007/BF02622099
author: Glyn Harman
author: Roger C. Baker
published in: Mathematische Zeitschrift
publication date: 1996-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174834
review text: Let \(R\) be a natural number and \({\mathbf x}\) an integer vector in \(t\) dimensions with \(|{\mathbf x}|\leq R^K\). There are positive constants \(c= c(K)\) and \(C_j= C_j (K, t)\) with the following property. Let \(C_1\leq Q\leq R^c\). Let \(\psi_1, \dots, \psi_t\) be positive numbers with \(\psi_i\leq C_2^{-1} (\log Q)^{-t}\) \((i=1,\dots,t)\), \(\psi_1 \dots \psi_t= Q^{-1}\). Then \(|{x_i} \pmod r|\leq r\psi_i\) (\(i=1,\dots, t)\) for some integer \(r\), \(R\leq r\leq 2RQ\). Examples are given to show that the range of \(r\) is essentially as short as possible. (Bombieri has an application to an effective sharpening of Liouville's theorem in diophantine approximation; we do not give details of this application, which are not available yet.) The key to the proof of our theorem is a `lattice method' of W. M. Schmidt.
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 861934
zbMATH Keywords: small remainder of a vector
zbMATH Keywords: lattice method of Schmidt
MaRDI profile type: Publication
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OpenAlex ID: W2019412010
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description: scientific article; zbMATH DE number 861934