Multiplication on uniform λ-Cantor sets

Jiangwen Gu¹ ; Kan Jiang¹ ; Lifeng Xi² ; Bing Zhao¹

¹ Ningbo University, Department of Mathematics
² Ningbo University, Department of Mathematics, and Hunan Normal University, School of Mathematics and Statistics

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 703-711

Résumé

  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.

Détail

Keywords: Self-similar set, uniform Cantor sets, arithmetic, Lebesgue measure