论文标题
厚度定理具有部分衍生物
Thickness theorems with partial derivatives
论文作者
论文摘要
在本文中，我们证明了一些具有部分导数的新厚度定理。我们提供一些申请。首先，我们给出一个简单的标准，可以判断两个缩放的cantor套件是否具有非空交叉路口。其次，我们在某些可检查条件下证明，具有正相似性比的任意自相似集的连续图像是封闭间隔，是封闭间隔或包含内部的有限结合。第三，我们证明了中间三分之一的cantor套装上的类似的Erdős-Straus猜想。最后，我们考虑了分形组上二芬太丁方程的解决方案。更具体地说，对于各种双磷剂方程，我们找不到某些相似集合的解决方案，而对于Fermat方程，这与著名的Fermat的最后一个定理相关，我们可以在许多自相似集合中找到许多解决方案。
In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under some checkable conditions that the continuous image of arbitrary self-similar sets with positive similarity ratios is a closed interval, a finite union of closed intervals or containing interior. Third, we prove an analogous Erdős-Straus conjecture on the middle-third Cantor set. Finally, we consider the solutions to the Diophantine equations on fractal sets. More specifically, for various Diophantine equations, we cannot find a solution on certain self-similar sets, whilst for the Fermat's equation, which is associated with the famous Fermat's last theorem, we can find infinitely many solutions on many self-similar sets.
