On topological properties of the set of maldistributed sequences

The real sequence (x ) is maldistributed if for any non-empty interval I, the set { ∈𝕅 : x ∈I} has upper asymptotic density 1. The main result of this note is that the set of all maldistributed real sequences is a residual set in the set of all real sequences (i.e., the maldistribution is a typical...

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Published inActa universitatis sapientiae. Mathematica Vol. 12; no. 2; pp. 272 - 279
Main Authors Bukor, József, Tóth, János T.
Format Journal Article
LanguageEnglish
Published Sciendo 01.11.2020
Scientia Publishing House
Subjects
54E52
Baire category
maldistributed sequence
weighted density
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Summary:The real sequence (x ) is maldistributed if for any non-empty interval I, the set { ∈𝕅 : x ∈I} has upper asymptotic density 1. The main result of this note is that the set of all maldistributed real sequences is a residual set in the set of all real sequences (i.e., the maldistribution is a typical property in the sense of Baire categories). We also generalize this result.
ISSN:2066-7752
2066-7752
DOI:10.2478/ausm-2020-0018