Generalized Cantor Set

Does that make sense? In summary, a generalized Cantor set E is defined by removing the middle r1 fraction of an interval, then removing the middle r2 fraction of the remaining 2 intervals, and so on. Starting with [0,1] and taking rn=1/5n, it can be shown that the material removed at the n-th stage has length < 1/5n, resulting in a total length removed of < 1/5 + 1/52 + 1/53 +... = 1/4. This means that the length of E is >3/4. The reasoning behind this is that at each stage, the FRACTION of intervals removed is multiplied by the length
  • #1
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"Given (rn), rn E (0,1), define a generalized Cantor set E by removing the middle r1 fraction of an interval, then remove the middle r2 fraction of the remaining 2 intervals, etc.

Start with [0,1]. Take rn=1/5n. Then the material removed at the n-th stage has length < 1/5n, so the total length removed is < 1/5 + 1/52 + 1/53 +... = 1/4
Thus the length of E is >3/4. "
=========================

I don't understand why the material removed at the n-th stage has length < 1/5n. How can we derive this? At the n-th stage, we are removing 2n-1 pieces, so don't we have to multiply that by 2n-1?
I sat down and thought about this for half an hour, but I still can't figure it out.

I hope someone can explain this! Thank you!
 
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  • #2
kingwinner said:
"Given (rn), rn E (0,1), define a generalized Cantor set E by removing the middle r1 fraction of an interval, then remove the middle r2 fraction of the remaining 2 intervals, etc.

Start with [0,1]. Take rn=1/5n. Then the material removed at the n-th stage has length < 1/5n, so the total length removed is < 1/5 + 1/52 + 1/53 +... = 1/4
Thus the length of E is >3/4. "
=========================

I don't understand why the material removed at the n-th stage has length < 1/5n. How can we derive this? At the n-th stage, we are removing 2n-1 pieces, so don't we have to multiply that by 2n-1?
I sat down and thought about this for half an hour, but I still can't figure it out.

I hope someone can explain this! Thank you!

It's true that you are removing [itex]2^{n-1}[/itex] intervals at stage [itex]n[/itex], but what is the LENGTH of each interval? It's not simply [itex]1/5^n[/itex], right? That is the FRACTION that you are removing from the preceding stage's intervals, so you have to multiply by the length of the preceding stage's intervals.

Since there are [itex]2^{n}[/itex] intervals remaining at the end of stage [itex]n[/itex], each one has length at MOST equal to [itex]1/2^{n}[/itex]. In reality the lengths are smaller because this assumes no material has been removed.

Therefore at stage [itex]n+1[/itex], you remove at most a length of [itex]2^n * (1/5^{n+1}) * (1/2^n) = 1/5^{n+1}[/itex]. (In reality you remove less.)
 
  • #3
jbunniii said:
Since there are [itex]2^{n}[/itex] intervals remaining at the end of stage [itex]n[/itex], each one has length at MOST equal to [itex]1/2^{n}[/itex]. In reality the lengths are smaller because this assumes no material has been removed.

I see what you're saying now: it is the FRACTION that you are removing from the preceding stage's intervals.

But I don't follow the reasoning here. Why are there 2n intervals remaining at the end of stage n?

And why each one has length at MOST equal to 1/2n? Can't it be the case that some of them has length less than 1/2n and some GREATER than 1/2n?

Thanks for explaining!
 
  • #4
kingwinner said:
I see what you're saying now: it is the FRACTION that you are removing from the preceding stage's intervals.

But I don't follow the reasoning here. Why are there 2n intervals remaining at the end of stage n?

You start with one interval. In stage 1, you remove the middle part of that interval so at the end of stage 1 there are [itex]2 = 2^1[/itex] intervals left.

In stage 2, you start with 2 intervals, and you remove the middle parts of each, so you are left with [itex]4 = 2^2[/itex] intervals at the end of stage 2. And so on.

And why each one has length at MOST equal to 1/2n? Can't it be the case that some of them has length less than 1/2n and some GREATER than 1/2n?

No, because the way the process works is symmetric - in a given stage, you are removing the MIDDLE part of each interval, and you are removing the same FRACTION of each interval. So all of the intervals at the end of a given stage have the same length. And that length cannot be more than [itex]1/2^n[/itex] because there are [itex]2^n[/itex] intervals and the original interval had length 1.
 
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What is a Generalized Cantor Set?

A Generalized Cantor Set is a mathematical set that is constructed by removing a middle portion of an interval and recursively repeating the process on the remaining intervals. It is a fractal with properties such as self-similarity and non-differentiability.

How is a Generalized Cantor Set different from a regular Cantor Set?

A Generalized Cantor Set is a generalization of the Cantor Set, which is constructed by removing the middle third of an interval. In a Generalized Cantor Set, the middle portion removed can be any fraction, not just one-third.

What are the applications of Generalized Cantor Sets?

Generalized Cantor Sets have applications in various areas of mathematics such as topology, number theory, and dynamical systems. They are also used in the study of chaos and fractals, and in the construction of self-similar structures.

Do Generalized Cantor Sets have a finite or infinite number of points?

A Generalized Cantor Set has an infinite number of points, as it is constructed by removing infinitely many intervals from the original interval. However, the set itself can have a finite or infinite measure depending on the fraction of the interval removed at each step.

Can a Generalized Cantor Set be visualized?

Yes, a Generalized Cantor Set can be visualized using a fractal plot or through the use of mathematical software. As the iterations continue, the set becomes more complex and exhibits self-similarity, making it a visually interesting object to study.

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