Multiplication on uniform λ-Cantor sets

Abstract

Let $C$ be the middle-third Cantor set. Define $C\ast C=\{x\ast y:x,y\in C\}$, where $\ast =+,-,\cdot ,÷$ (when $\ast =÷$, we assume $y\ne 0$). Steinhaus [17] proved in 1917 that $C-C=[-1,1]$, $C+C=[0,2]$. In 2019, Athreya, Reznick and Tyson [1] proved that $C÷C=\bigcup _{n=-\mathrm{\infty }}^{\mathrm{\infty }}[3^{-n}{\displaystyle \frac{2}{3}},3^{-n}{\displaystyle \frac{3}{2}}]\cup \{0\}$. 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.