论文标题
权力之和中间第三康托尔集的元素的产品和产品总和
Sums of powers, and products of elements of the middle third Cantor set
论文作者
论文摘要
最近在中间第三cantor集（或IT的变体）的某些总和和产物中找到的量度和寻找开放间隔的问题，最近引起了人们的极大兴趣。 ASTEL概述了一个可以解决这些问题的广泛一般框架。 Athreya，Reznick和Tyson的一篇文章最近考虑了寻找中部第三康托尔集产品产品的量度的问题。 ASTELS的方法适用于考虑广义的对数Cantor套件，而Athreya，Reznick和Tyson的方法在处理$ M $ M $'powers的总和很难处理时很难。有了一种新的基本动力学技术，我们可以以令人满意的方式处理总和和产品的问题，并且证明与基本证据的复杂性水平相同，证明了这一事实，即中间第三个Cantor Set的两个副本为$ [0,2] $。此外，对一个中央康托尔（Central Cantor）的相交问题进行了一些观察，该问题设置了另一个中央cantor的仿射图像。
The questions of the measure and finding open intervals in certain sets of sums and products of elements of the middle third Cantor set (or a variant of it), have generated considerable interest recently. A broad general framework that makes it possible to deal with these questions was outlined by Astels. The question of finding the measure of the product of the middle third Cantor set was considered recently in an article by Athreya, Reznick and Tyson. Astels' methods apply to product sets upon considering a generalized logarithmic Cantor set, while Athreya, Reznick and Tyson's methods become difficult in dealing with sums or products of $m$'th powers as $m$ becomes large. With a new elementary dynamical technique, we can deal with both the questions of sums and products in a satisfactory way, and the proofs are of the same level of complexity as that of an elementary proof of the fact that the sum of two copies of the middle third Cantor set is $[0,2]$. Further, some observations are made on the question of intersections of one central Cantor set with an affine image of another central Cantor set both with arbitrary parameters.
