Multiplication on 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.