Multiplication on uniform λ-Cantor sets
| Autor: | Lifeng Xi, Jiangwen Gu, Bing Zhao, Kan Jiang |
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| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Annales Fennici Mathematici |
| ISSN: | 2737-114X 2737-0690 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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