Multiplication on uniform $\lambda$-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 \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C÷C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] 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.

9 pages

Cited by in corpus (3)

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