Strong characterizing sequences in simultaneous Diophantine approximation.
type: Item
instance of: scholarly article
title: Strong characterizing sequences in simultaneous Diophantine approximation.
zbMATH Open document ID: 1058.11047
DOI: 10.1016/S0022-314X(02)00068-9
author: András Biró
author: Vera T. Sós
published in: Journal of Number Theory
publication date: 2003-05-25T00:00:00Z
review text: In this paper, it is proved that: if \(1,\alpha_1,\dots, \alpha_t\in\mathbb{R}\) are linearly independent over the rationals, there is a subset \(A\subset\mathbb{N}\), \(| A|=\infty\), such that \(\sum_{n\in A}\| n\beta\|\) is finite if and only if \(\beta\in G\), the group generated by \(1,\alpha_1,\dots, \alpha_t\). Moreover, if \(\beta\not\in G\), then we even have \(\liminf_{n\in A,n\to+\infty}\| n\beta\|> 0\). This strengthens a theorem proved in [\textit{A. Biró}, \textit{J.-M. Deshouillers}, and \textit{V. T. Sós}, Stud. Sci. Math. Hung. 38, 97--113 (2001; Zbl 1006.11038)]. The proof uses both ideas from this paper and Freiman's theory (in the form of a structural lemma on Bohr sets).
reviewed by: Alain Plagne
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11P70
zbMATH DE Number: 1915514
zbMATH Keywords: Characterizing sequences
zbMATH Keywords: Bohr sets
zbMATH Keywords: Freiman's theorem
MaRDI profile type: Publication
cites work: Good approximation and characterization of subgroups of R = Z
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cites work: Q5686108
cites work: Generalized arithmetical progressions and sumsets
cites work: Q3215325
description: scientific article; zbMATH DE number 1915514