On series of translates of positive functions. II
type: Item
instance of: scholarly article
title: On series of translates of positive functions. II
zbMATH Open document ID: 1025.28001
DOI: 10.1016/S0019-3577(01)80013-6
author: Zoltán Buczolich
author: R. Daniel Mauldin
published in: Indagationes Mathematicae. New Series
publication date: 2002-07-11T00:00:00Z
review text: This paper is a continuation of the earlier paper [\textit{Z. Buczolich, J.-P. Kahane} and \textit{R. D. Mauldin}, Acta Math. Hung. 93, 171-188 (2001; Zbl 1006.28005)]. It gives a detailed study of the set of convergence of series of the form \(\sum_{\lambda \in \Lambda} f(x+\lambda)\) for a nonnegative measurable function \(f.\) From the authors' summary: ``Given a sequence of natural numbers \(n_1 < n_2 < \ldots \) we are interested in sets \(\Lambda\) of the form \(\bigcup^ {(x)}_{k=1} \alpha ^ k \mathbb{Z} \cap [n_k, n_{k+1}),\) where \(0<\alpha < 1.\) In case \(\alpha = 1/q,\) where \(q>1\) is an integer, there is a zero-one law showing that for every measurable \(f : \mathbb{R} \rightarrow \mathbb{R}^ +\) the above sum either converges almost everywhere or diverges almost everywhere. However, for any other value of \(\alpha \in (0,1)\) there is no such zero-one law.'' The main work is in constructing explicitely measurable sets \(A \subset \mathbb{R}\) such that \(f=1_A\) serves as counterexample.
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 60F20
zbMATH DE Number: 1766473
zbMATH Keywords: convergence of series
zbMATH Keywords: measurable function
zbMATH Keywords: zero-one law
zbMATH Keywords: measurable sets
reviewed by: Heinrich von Weizsäcker
MaRDI profile type: Publication
cites work: On series of translates of positive functions
cites work: Sur les séries de translatées de fonctions positives
cites work: On the convergence of for measurable functions
cites work: Q3885825
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cites work: On Khinchin's Conjecture about Strong Uniform Distribution
full work available at URL: s0019-3577(01)80013-6
OpenAlex ID: W2884678507
description: scientific article; zbMATH DE number 1766473