Multiplication on uniform $\lambda$-Cantor sets

TL;DR

This paper investigates the topological and measure-theoretic properties of the product set of the middle-third Cantor set and extends these results to uniform λ-Cantor sets, revealing new insights into their structure.

Contribution

It provides a detailed description of the topological structure and Lebesgue measure of the product of the Cantor set and generalizes these findings to uniform λ-Cantor sets.

Findings

Characterization of the topological structure of C·C

Determination of Lebesgue measure of C·C

Extension of results to uniform λ-Cantor sets

Abstract

Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,\div$ (when $*=\div$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C\div C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] In this paper, we give a description of the topological structure and Lebesgue measure of $C\cdot C$. We indeed obtain corresponding results on the uniform $\lambda$-Cantor sets.