Cantor set arithmetic

TL;DR

This paper explores the algebraic and measure-theoretic properties of the Cantor set under multiplication and division, revealing that every number in [0,1] can be expressed as a product of Cantor set elements and analyzing the structure of their quotients.

Contribution

It demonstrates that any number in [0,1] can be written as a product of three Cantor set elements and characterizes the measure and structure of sets formed by products and quotients of Cantor set elements.

Findings

Every element in [0,1] can be written as x^2 y with x,y in C.

The set of products xy with x,y in C has measure between 17/21 and 8/9.

The structure of the quotient set C/C is described in detail.

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 $f(x,y) = x/y$.