@article{3628d4c14ef14a40bfa6582e9004c1ad,

title = "Cantor Set Arithmetic",

abstract = "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.",

keywords = "MSC: Primary 28A80, Secondary 11K55",

author = "Athreya, {Jayadev S.} and Bruce Reznick and Tyson, {Jeremy T.}",

note = "Funding Information: ACKNOWLEDGMENTS. The authors wish to thank the referees and editors for their rapid, sympathetic, and extremely useful suggestions for improving the manuscript. We also thank Jeff Shallit and Yuki Takahashi for providing additional references. B.R. wants to thank Professor W. A. J. Luxemburg (1929–2018) for his course Math 108 at Caltech in 1970–1971, which introduced him to the beauties of analysis. J.S.A. was supported by NSF CAREER grant DMS-1559860 and NSF grants DMS-1069153, DMS-1107452, DMS-1107263, and DMS-1107367. B.R. was supported by Simons Collaboration Grant 280987. J.T.T. was supported by NSF grants DMS-1201875 and DMS-1600650 and Simons Collaboration Grant 353627. Publisher Copyright: {\textcopyright} 2018, {\textcopyright} THE MATHEMATICAL ASSOCIATION OF AMERICA.",

year = "2019",

month = jan,

day = "2",

doi = "10.1080/00029890.2019.1528121",

language = "English (US)",

volume = "126",

pages = "4--17",

journal = "American Mathematical Monthly",

issn = "0002-9890",

publisher = "Mathematical Association of America",

number = "1",

}