A nonlinear version of the Newhouse thickness theorem

TL;DR

This paper extends Newhouse's thickness theorem to nonlinear functions of multiple Cantor sets, establishing conditions under which their images form intervals, with applications to continued fractions and solving related mathematical questions.

Contribution

It generalizes the classical thickness theorem to nonlinear functions of multiple Cantor sets, providing new conditions for their images to be intervals and applying these results to continued fractions.

Findings

The nonlinear thickness theorem guarantees interval images under specified derivative bounds.

Applications include answering open questions and deriving identities related to continued fractions.

The results connect Cantor set properties with nonlinear transformations in real analysis.

Abstract

Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $\tau(C_1)\cdot \tau(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $\tau(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset \mathbb{R}, i=1,\cdots, d$, be some Cantor sets (perfect and nowhere dense) with convex hull $[0,1]$. Suppose $f(x_1,\cdots, x_{d-1},z)\in \mathcal{C}^1$ is a continuous function defined on $\mathbb{R}^d$. Denote the continuous image of $f$ by $$f(K_1,\cdots, K_d)=\{f(x_1, \cdots x_{d-1},z):x_i\in K_i,z\in K_d, 1\leq i\leq d-1\}.$$ If for any $(x_1, \cdots, x_{d-1},z)\in [0,1]^d$, we have $$(\tau(K_i))^{-1}\leq \left|\dfrac{\partial_{x_i} f}{\partial_z f}\right|\leq \tau(K_d),1\leq i\leq d-1$$ then $f(K_1,\cdots, K_d)$ is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.