幂和，以及中间三分之一 Cantor 集元素的乘积
Sums of powers, and products of elements of the middle third Cantor set
AI总结：
本文提出一种新的初等动力学技术，统一处理中间三分之一 Cantor 集元素的幂和与幂积的测度及开区间问题，克服了已有方法在幂次增大时的困难。
摘要：
关于中间三分之一 Cantor 集（或其变体）元素的某些和集与积集的测度问题以及在其中寻找开区间的问题，近来引起了相当大的兴趣。Astels 概述了一个能够处理这些问题的广泛通用框架。Athreya、Reznick 和 Tyson 最近在一篇论文中研究了求中间三分之一 Cantor 集乘积测度的问题。Astels 的方法通过考虑广义对数 Cantor 集而适用于乘积集，而 Athreya、Reznick 和 Tyson 的方法在处理 m 次幂的和或积时，随着 m 变大而变得困难。借助一种新的初等动力学技术，我们能够以令人满意的方式同时处理和与积这两类问题，并且证明的复杂程度与证明中间三分之一 Cantor 集的两个副本之和为 [0,2] 这一事实的初等证明处于同一水平。此外，本文还就一个中心 Cantor 集与另一个均具有任意参数的中心 Cantor 集的仿射像之间的交的问题给出了若干观察。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
