Arithmetic sums of cantor sets  (Page 24/6)

Introduction

The primary objective of this study is to give stronger characterizations for the arithmetic sum of two Cantor sets. We limited ourselves to the case of sums of two different mid- $\alpha$ Cantor sets. It is a straightforward question but still unsolved in the majority of cases. It was first posed by J. Palis and F. Takens in the context of Dynamical Systems [link] . It also has applications in Number Theory [link] , Physics [link] , and is interesting from a purely topological perspective.

In this module, we will give preliminary definitions, provide known results, and then present the results from our study.

Preliminaries

Mid- $\alpha$ Cantor sets

A Cantor Set $C$ is a set with the following properties:

• $C$ is non-empty.
• $C$ is compact.
• $C$ is perfect.
• $C$ is totally disconnected.

These properties imply that $C$ is uncountable. The canonical example is the Cantor ternary set $T$ , constructed in the following way:

1. Take $T_0=\left[0,,,1\right]$ to be the unit interval, and remove the "middle third" from $T_0$ to get $T_1=\left[0,,,\frac{1}{3}\right]\cup \left[\frac{2}{3},,,1\right]$ ( [link] ).
2. Remove again the "middle thirds" from the two remaining connected components of $T_1$ to get $T_2=\left[0,,,\frac{1}{9}\right]\cup \left[\frac{2}{9},,,\frac{1}{3}\right]\cup \left[\frac{2}{3},,,\frac{7}{9}\right]\cup \left[\frac{8}{9},,,1\right]$ ( [link] ).
3. Repeat this process. The desired Cantor ternary set is $T=\bigcap _{j=0}^{\infty }{T}_j$ ( [link] ).

The construction of the Cantor ternary set may be generalized slightly by giving ourselves a varying parameter $\alpha$ . In this case, we had $\alpha =\frac{1}{3}$ . For example, if we take $\alpha =\frac{1}{2}$ , then we remove the "middle halves" of intervals at each stage.

These so-called mid- $\alpha$ Cantor sets are the building blocks for our study. However, it becomes more convenient to reference them in terms of $\lambda =\frac{1}{2}\left(1,-,\alpha \right)$ , i.e. the lengths of the remaining intervals in the first stage of the construction. For a given $\lambda$ , as done by Mendes and Oliveira, we denote the corresponding mid- $\left(1,-,2,\lambda \right)$ Cantor set as $C\left(\lambda \right)$ [link] .

We may represent an arbitrary point $x\in C\left(\lambda \right)$ in the form

With this notation in mind, we think of $\lambda$ as a scaling factor and $A\left(\lambda \right)$ as a set of offsets in the sense that $C\left(\lambda \right)$ consists of two copies of itself, scaled by $\lambda$ , and translated by the elements of $A\left(\lambda \right)$ . That is,

where $\lambda ·C\left(\lambda \right)+\left(1,-,\lambda \right)$ is interpreted as

Homogeneous cantor sets

We may further generalize the construction of the Cantor sets $C\left(\lambda \right)$ to allow for more possibilities for the set of offsets $A=\left\{a_0,,,a_1,,,\cdots ,,,a_k\right\}$ for some $k\ge 1$ . In this case, we have a homogeneous Cantor set $C\left(\lambda ,,,A\right)$ , which can be represented (with a slight abuse of notation) as