TY - JOUR AU - Gu, Jiangwen AU - Jiang, Kan AU - Xi, Lifeng AU - Zhao, Bing PY - 2021/08/02 Y2 - 2024/03/29 TI - Multiplication on uniform λ-Cantor sets JF - Annales Fennici Mathematici JA - Ann. Fenn. Math. VL - 46 IS - 2 SE - Articles DO - UR - https://afm.journal.fi/article/view/110571 SP - 703-711 AB - <p>&nbsp;</p><pre>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 \(y eq0\)). Steinhaus [17] proved in 1917 that</pre><pre>\(C-C=[-1,1]\), \(C+C=[0,2]\).</pre><pre>In 2019, Athreya, Reznick and Tyson [1] proved that</pre><pre>\(C\div C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] \cup \{0\}\).</pre><pre>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.</pre> ER -