Cantor Set Arithmetic

Every element u of [0,1] can be written in the form u = x2y, where x, y are elements of the Cantor set C. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand the set of real numbers v that can be written in the form v = xy with...

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Bibliographic Details
Published inThe American mathematical monthly Vol. 126; no. 1; p. 4
Main Authors Athreya, Jayadev S, Reznick, Bruce, Tyson, Jeremy T
Format Journal Article
LanguageEnglish
Published Washington Taylor & Francis Ltd 01.01.2019
Subjects
Mathematical analysis
Mathematical functions
Real numbers
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Summary:Every element u of [0,1] can be written in the form u = x2y, where x, y are elements of the Cantor set C. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand the set of real numbers v that can be written in the form v = xy with x and y in C is a closed subset of [0,1] with Lebesgue measure strictly between 17/21 and 8/9. We also describe the structure of the quotient of C by itself, that is, the image of C × (C \ {0}) under the function f (x, y) = x/y.
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2018.1528121