Cantor set arithmetic

Abstract

Every element $u$ of $[0,1]$ can be written in the form $u=x^2y$, 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 $\tfrac{17}{21}$ and $\tfrac89$. We also describe the structure of the quotient of $C$ by itself, that is, the image of $C\times (C \setminus \{0\})$ under the function $f(x,y) = x/y$.