# Arithmetic sums of cantor sets

 Page 1 / 6
This report summarizes work done as part of Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. This module explores the topolgical properties of the sum of two central homogeneous Cantor sets and presents stronger sufficient conditions for these properties. This study was led by Dr. Danijela Damjanovic and Dr. David Damanik.

## 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.

## 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

$x=\sum _{n=0}^{\infty }{\alpha }_{n}{\lambda }^{n}\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{1.em}{0ex}}{\alpha }_{n}\in A\left(\lambda \right)=\left\{0,,,1,-,\lambda \right\}\phantom{\rule{3.33333pt}{0ex}}\forall n.$

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,

$C\left(\lambda \right)=\lambda ·C\left(\lambda \right)\cup \left[\lambda ,·,C,\left(\lambda \right),+,\left(1,-,\lambda \right)\right]$

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

$\left\{\lambda ,·,x,+,\left(1,-,\lambda \right),\mid ,x,\in ,C,\left(\lambda \right)\right\}.$

## 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

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

 By Nick Swain By