Cantor Set Arithmetic

Jayadev S. Athreya, Bruce Reznick, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)4-17
Number of pages14
JournalAmerican Mathematical Monthly
Issue number1
StatePublished - Jan 2 2019


  • MSC: Primary 28A80
  • Secondary 11K55

ASJC Scopus subject areas

  • General Mathematics


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