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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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
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