Thickness theorems with partial derivatives

TL;DR

This paper establishes new thickness theorems involving partial derivatives, providing criteria for intersections of Cantor sets, images of self-similar sets, and solutions to Diophantine equations on fractals.

Contribution

It introduces novel thickness theorems with partial derivatives and applies them to Cantor sets, self-similar sets, and Diophantine equations, extending classical results.

Findings

Criteria for intersection of scaled Cantor sets

Images of self-similar sets can be intervals or unions

Existence of solutions to Fermat's equation on fractals

Abstract

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 Erd\H{o}s-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.