On series of translates of positive functions
type: Item
instance of: scholarly article
title: On series of translates of positive functions
zbMATH Open document ID: 1006.28005
DOI: 10.1023/A:1013904610368
author: Zoltán Buczolich
author: Jean-Pierre Kahane
author: R. Daniel Mauldin
published in: Acta Mathematica Hungarica
publication date: 2003-02-18T00:00:00Z
review text: Let us consider the collection \({\mathcal D}\) of all discrete infinite sets of nonnegative real numbers. For \( \Lambda \in {\mathcal D} \) and a nonnegative Lebesgue measurable function \( f : {\mathbb{R}} \to {\mathbb R}^{+} \) let \( C(f,\Lambda) = \{ x \in {\mathbb R} : \sum_{\lambda \in \Lambda} f(x+\lambda) < +\infty\} \). The collection \({\mathcal D}\) is divided into two types. Type 1 consists of \( \Lambda \in {\mathcal D} \) such that, for every admissible \(f\), either \( C(f,\Lambda) \) is a null set or \( {\mathbb R} \setminus C(f,\Lambda) \) is a null set (in the sense of Lebesgue measure). Type 2 contains the rest of \({\mathcal D}\). Classes of examples for sets of both types are enumerated. For instance, it is demonstrated that \(\Lambda\) is of type 2 if it is asymptotically lacunary or it is asymptotically dense and exhibits asymptotically large \({\mathbb Q}\)-independent sets. It is also established that sets of type 2 form a dense open subset of \({\mathcal D}\), while sets of type 1 form a closed nowhere dense subset of \({\mathcal D}\) in the box topology.
reviewed by: Zoltán Boros
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 60F20
Mathematics Subject Classification ID: 42A75
zbMATH DE Number: 1870466
zbMATH Keywords: asymptotically dense
zbMATH Keywords: asymptotically lacunary
zbMATH Keywords: Borel-Cantelli lemma
zbMATH Keywords: \(\mathbb Q\)-linear subspace
zbMATH Keywords: periodic and almost periodic functions
MaRDI profile type: Publication
full work available at URL: a:1013904610368
OpenAlex ID: W2962706650
description: scientific article; zbMATH DE number 1870466