Suites de longueur minimale associées à un ensemble normal donné. (Sequences of minimal length associated with a given normal set)
type: Item
instance of: scholarly article
title: Suites de longueur minimale associées à un ensemble normal donné. (Sequences of minimal length associated with a given normal set)
zbMATH Open document ID: 0682.10045
DOI: 10.1007/BF02787225
author: Jean-Pierre Borel
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A set \({\mathcal A}\subseteq {\mathbb{R}}\) is called b-normal if there exists a bounded sequence \(\Lambda =(\lambda_ 1,\lambda_ 2,...)\) in \({\mathbb{R}}\) such that \({\mathcal A}\) is the set \({\mathcal B}(\Lambda)\) of all x for which \((x\lambda_ n)\) is u.d. mod 1. The author investigates the structure of such sets \({\mathcal A}\). In particular, he establishes relations between the ``minimal bound'' \[ M({\mathcal A}):=\inf \{M\geq 0:\quad \exists \Lambda,\quad {\mathcal A}={\mathcal B}(\Lambda),\quad 0\leq \lambda_ n\leq M\quad \forall n\} \] and various densities of \({\mathcal A}\). Especially interesting results are obtained in the case \({\mathcal A}\subseteq {\mathbb{Z}}\). E.g., any set \({\mathcal A}\) of the form \(\cup^{k}_{i=1}\gamma_ i({\mathbb{Z}}\setminus \{0\})\), with natural numbers \(\gamma_ i\), is shown to be b-normal and to satisfy \[ M({\mathcal A})\geq d({\mathcal A}):=\lim_{m\to \infty}(1/2m)\quad \#\{a\in {\mathcal A}:\quad | a| <m\}, \] with equality holding iff all coefficients of the polynomial \(P(t)=\prod_{a\in \bar A}(t-\zeta^ a_ m)\) are positive, where \(m=lcm\{\gamma_ 1,...,\gamma_ k\}\), \(\zeta_ m=e^{2\pi i/m}\), \(\bar A={\mathcal A}\cap \{1,2,...,m-1\}\).
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11K16
zbMATH DE Number: 4116647
zbMATH Keywords: normal sets
zbMATH Keywords: uniform distribution
zbMATH Keywords: Fourier transforms
zbMATH Keywords: minimal bound
zbMATH Keywords: densities
reviewed by: Bodo Volkmann
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02787225
OpenAlex ID: W1998197363
description: scientific article; zbMATH DE number 4116647