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For each θ 1 there is a unique λ θ λ 1 θ , λ 2 θ such that C λ + C λ θ = 0 , 2 if and only if λ λ θ . Moreover,

  • if θ is irrational, then λ θ = λ 2 θ ;
  • if θ = n or θ = 1 + 1 n where n is a positive integer, then λ θ = λ 1 θ .

The lines λ 1 θ and λ 2 θ can be seen in [link] as the blue and red lines, respectively.

A partial to this result was given by Cabrelli, Hare, and Molter in 2002, regarding the possibilities that λ θ = λ 1 θ or λ θ = λ 2 θ [link] .

Theorem 3.5 (Cabrelli, Hare, and Molter 2002) Using the notation from above, we have the following results:

  • λ θ = λ 1 θ if and only if θ = n or θ = 1 + 1 n for some positive integer n ;
  • Let θ = 1 + p q where gcd p , q = 1 and p = 1 , 2 , , 8 or let θ = n + 1 2 with n N . Then, λ θ < λ 2 θ .

Theorem 3.4 proved that (i) is sufficient, and this theorem proves that it is also necessary. As for (ii) , we now know that there are some rational θ , but not necessarily all, for which λ θ must lie strictly below λ 2 θ . It is conjectured, but not known, that this is true for all rational θ .

A summary of the known results presented in this section is given below in [link] .

As mentioned in "Sums of Mid-α Cantor Sets" , we now know that C 1 3 + C 1 3 = 0 , 2 because 1 3 λ 1 = 1 3 and that C 1 5 + C 1 5 is a Cantor set because the point 1 , 1 5 in the θ , λ -plane lies in Region I.

A visualization of the θ , λ -plane. Region I indicates the area where H D C λ + H D C λ θ < 1 , thus C λ + C λ θ must be a Cantor set with zero Lebesgue measure by Lemma 3.2. By Theorem 3.3, we have Hausdorff dimension 1 for irrational θ in Regions II and III. By Theorems 3.4 and 3.5, we have the full interval 0 , 2 in Region III and, for some θ , in Region II as well. However, the full interval never occurs below the blue line.

Results from the study

In our study, we decided to focus on rational θ . We found a nice way to write the sum set C λ + C λ θ when θ = p q , where gcd p , q = 1 , that is similar to the notation used for homogeneous Cantor sets. Sets of this form may be homogeneous Cantor sets, but in many cases they are not. We will present this process first for θ = 2 and then for general rational θ .

The idea behind this process is to write the two mid- α Cantor sets as homogeneous Cantor sets with the same scaling factor. For θ = 2 , remember that we may write C λ as a homogeneous Cantor set with scaling factor λ , as in

C λ = n = 0 α n λ n , α n A λ = 0 , 1 - λ .

The second stage of the construction of C λ , as described in "Mid-α Cantor Sets" , is the union of 4 intervals, each of length λ 2 , starting at the points 0 , λ 1 - λ , 1 - λ , and 1 + λ 1 - λ . With this in mind, we can rewrite C λ as a homogeneous Cantor set with scaling factor λ 2 and 4 offsets. That is,

C λ = n = 0 α n λ 2 n , α n A = 0 , λ 1 - λ , 1 - λ , 1 - λ 2 .

One might notice here that A = A λ + λ · A λ . And since θ = 2 , we have also that

C λ 2 = n = 0 β n λ 2 n , β n B = 0 , 1 - λ 2 .

Now, we can take the sum

C λ + C λ 2 = n = 0 α n + β n λ 2 n , α n A = 0 , λ 1 - λ , 1 - λ , 1 - λ 2 , β n B = 0 , 1 - λ 2 .

Or equivalently,

C λ + C λ 2 = n = 0 γ n λ 2 n , γ n A + B

where A + B is considered as the normal set-wise sum. In this case, we have

A + B = 0 , λ 1 - λ , 1 - λ , 1 + λ 1 - λ , 1 + 2 λ 1 - λ , 2 + λ 1 - λ , 2 + 2 λ 1 - λ .

This sum C λ + C λ 2 is contained in the interval 0 , 2 . However, due to our notation, it is much simpler and more convenient to consider a scaled-down copy of this set so that it is contained in the interval 0 , 1 . We do this by halving the offsets. This does not change any properties of the set. Also, each offset contains a factor of ( 1 - λ ) . This is not an accident, and we will use it here to condense the notation, writing

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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