On Kakeya Conditions for Achievement Sets

We prove that for each infinite subset C of N there exists a sequence ( x n ) such that { n : x n > r n } = C and the achievement set A ( x n ) is a Cantor set. Moreover, we show that it is possible to construct a sequence ( x n ) such that the set { n : x n > r n } has asymptotic density α fo...

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Bibliographic Details
Published inResultate der Mathematik Vol. 76; no. 4
Main Authors Marchwicki, Jacek, Miska, Piotr
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2021
Subjects
Mathematics
Mathematics and Statistics
absolutely convergent series
Cantor set
Achievement set
set of subsums
purely atomic measure
Cantorval
Primary: 40A05
Secondary: 11K31
asymptotic density
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Summary:We prove that for each infinite subset C of N there exists a sequence ( x n ) such that { n : x n > r n } = C and the achievement set A ( x n ) is a Cantor set. Moreover, we show that it is possible to construct a sequence ( x n ) such that the set { n : x n > r n } has asymptotic density α for each α ∈ [ 0 , 1 ) and A ( x n ) is a Cantorval.
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-021-01479-2