Multiplication on uniform λ-Cantor sets

Abstract

Let \(C\) be the middle-third Cantor set. Define \(C*C=\{x*y\colon x,y\in C\}\), where \(*=+,-,\cdot,\div\) (when \(*=\div\), we assume \(y\neq0\)). Steinhaus [17] proved in 1917 that \(C-C=[-1,1]\), \(C+C=[0,2]\). In 2019, Athreya, Reznick and Tyson [1] proved that \(C\div C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] \cup \{0\}\). 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.