The Cantor set

[SLD]

The cantor set is really cool. Take the line from 0 to 1, inclusive. Take out the middle third, and then take out the middle third of the two remaining thirds, and keep going to infinity. Here is a picture of the first six iterations.:



So what’s left? Nothing and infinity!

If you measure the “length” of the remaining pieces, it is zero. For each iteration, n, the remaining length of all the pieces, is (2/3)^n. But if you sum up the remaining pieces, then the answer is infinity. That’s because despite the infinite process, there are still no isolated points.

You can also do it in three or more dimensions. Cantor dust, or a Cantor sponge.

 

[Swammerdami]

A few years before Cantor published his Cantor Set, Henry Smith wrote on the topic and also introduced what is now called the  Smith–Volterra–Cantor set. This set, like the Cantor set, has no intervals but differs from Cantor's by making progressively smaller deletions and therefore having positive measure (total length = 1/2). "This makes the Smith–Volterra–Cantor set an example of a closed set whose boundary has positive Lebesgue measure." It leads to the paradoxical  Volterra's function.

 

[Jarhyn]

A few years before Cantor published his Cantor Set, Henry Smith wrote on the topic and also introduced what is now called the  Smith–Volterra–Cantor set. This set, like the Cantor set, has no intervals but differs from Cantor's by making progressively smaller deletions and therefore having positive measure (total length = 1/2). "This makes the Smith–Volterra–Cantor set an example of a closed set whose boundary has positive Lebesgue measure." It leads to the paradoxical  Volterra's function.

If I may ask, what is the breaking point of growth rate wherein the reduction of the reduction leads to convergence on a real number, and where the system converges instead to 0 measure, despite having infinite members?

It seems something related to e and geometric growth and ln in my head?

Edit: it just seems really useful in thinking about a function you want to have move from a flat 0 to a real number, wiggle around there as a real number, and then stay at a flat 0 and be well behaved there again.

 

[Swammerdami]

If I may ask, what is the breaking point of growth rate wherein the reduction of the reduction leads to convergence on a real number, and where the system converges instead to 0 measure, despite having infinite members?

Short answer: The Smith-Volterra-Cantor set has zero measure when 1/3 is substituted for the 1/4 parameter.

I think there are various ways a class of "Fat Cantor Sets" MIGHT be defined, but the actual definition for SVC sets leads to great simplicity.
With initial length 1, a = 1/4 of length from the single segment is removed in the first iteration; then 1/16 of length from each of two segments in the 2nd iteration; then 1/64 from each of four segments and so on. Those removals are absolute values, NOT portions.

The total removal is 1*1/4 + 2*1/16 + 4*1/64 + 8*1/256 + ... = 1/4 + 1/8 + 1/16 + ... = 1/2
More generally, total removal is a¹ + 2a² + 4a³ + 8a⁴ + ... = a/(1-2a)
So, 0 is removed when a=0; 1/4 when a = 1/6; 1/3 when a = 1/5; 1/2 when a = 1/4; 1 when a = 1/3. The construction fails when a > 1/3.
This info is shown (in a different format) at  Smith–Volterra–Cantor_set#Other_fat_Cantor_sets.

 

[Jarhyn]

If I may ask, what is the breaking point of growth rate wherein the reduction of the reduction leads to convergence on a real number, and where the system converges instead to 0 measure, despite having infinite members?

Short answer: The Smith-Volterra-Cantor set has zero measure when 1/3 is substituted for the 1/4 parameter.

I think there are various ways a class of "Fat Cantor Sets" MIGHT be defined, but the actual definition for SVC sets leads to great simplicity.
With initial length 1, a = 1/4 of length from the single segment is removed in the first iteration; then 1/16 of length from each of two segments in the 2nd iteration; then 1/64 from each of four segments and so on. Those removals are absolute values, NOT portions.

The total removal is 1*1/4 + 2*1/16 + 4*1/64 + 8*1/256 + ... = 1/4 + 1/8 + 1/16 + ... = 1/2
More generally, total removal is a¹ + 2a² + 4a³ + 8a⁴ + ... = a/(1-2a)
So, 0 is removed when a=0; 1/4 when a = 1/6; 1/3 when a = 1/5; 1/2 when a = 1/4; 1 when a = 1/3. The construction fails when a > 1/3.
This info is shown (in a different format) at  Smith–Volterra–Cantor_set#Other_fat_Cantor_sets.

What I'm getting at is that it's really no different a question than let's say,

I start with a thing of size 1; if every time I remove one half of what remains, this tends towards 0.

If, instead, I remove a half, quarter, eighth, and so on, I find myself approaching a value of 0.

As noted, the cantor set removes thirds of each remainder and still approaches 0 asymptotically as iterations increase.

I could do it the cantor way of removing those inner segments, but I would get the same measure just progressively removing that whole fraction, essentially viewing those infinite segments of the cantor set as "compressed to the side", and then just viewing it as a vessel progressively losing some ratio of itself until it is "practically empty".

Logically, if I removed less than a half, less than a quarter, less than an eighth, and so on, I would instead end up losing "real estate" at a rate that wouldn't approach 0, but would approach the remainder of whatever difference between 1/2 and the actual amount removed is, plus the difference between whatever and 1/4, plus the difference between that and 1/8, and this fractional amount would over time "cut" the loss?

My intuition tells me that the remainder of the subtraction would be equal to the sum of the addition of the difference of removal each time, which would itself converge to a small number, but I'm not sure this is the case given how weird infinity gets.

If the same happens in Cantor that measure ends up approaching 0 with excised thirds, if each time I removed less than that third, and then less than that third of two thirds that would be the original remainder, again I would find myself approaching a nonzero number.

I guess my question is, what is the identity of that element of the rate?

 

[Swammerdami]

Let me pose a question similar to yours, but perhaps more concrete. I do NOT know the answer to either this question or yours. :-(

For brevity let
(a,b,c,d,...) denote the remainder after removing portion a, then portion b of what's left (i.e. b*(1-a), then portion c of what's then left and so on.
We wonder whether that process applied to an initial unit will eventually result in Zero (asymptotically). Here are some examples:

• (1/3, 1/3, 1/3, 1/3, ... ) --> Zero // this is the Cantor set
• (1/1000, 1/1000, 1/1000, 1/1000, ...) --> Zero // Although converging slowly it can have no non-zero asymptote)
• (1/2, 1/3, 1/4, 1/5, 1/6, ...) --> Zero // Somewhat harder to prove
• (1/4, 1/4², 1/4³, 1/4⁴, ...) --> Non-zero // (I think) because it empties slower than the SVC set.

But by now, I am lost. I'm not even sure of the final line.

What is sought here, and perhaps also by Jarhyn IIUC, is a process in this form which is "critical", i.e. which fails to converge to Zero but ALMOST succeeds.

I don't have an answer.

 

[Jarhyn]

Let me pose a question similar to yours, but perhaps more concrete. I do NOT know the answer to either this question or yours. :-(

For brevity let
(a,b,c,d,...) denote the remainder after removing portion a, then portion b of what's left (i.e. b*(1-a), then portion c of what's then left and so on.
We wonder whether that process applied to an initial unit will eventually result in Zero (asymptotically). Here are some examples:

• (1/3, 1/3, 1/3, 1/3, ... ) --> Zero // this is the Cantor set
• (1/1000, 1/1000, 1/1000, 1/1000, ...) --> Zero // Although converging slowly it can have no non-zero asymptote)
• (1/2, 1/3, 1/4, 1/5, 1/6, ...) --> Zero // Somewhat harder to prove
• (1/4, 1/4², 1/4³, 1/4⁴, ...) --> Non-zero // (I think) because it empties slower than the SVC set.

But by now, I am lost. I'm not even sure of the final line.

What is sought here, and perhaps also by Jarhyn IIUC, is a process in this form which is "critical", i.e. which fails to converge to Zero but ALMOST succeeds.

I don't have an answer.

BUT yeah, I know I've observed series which converge on numbers like 1/sqrt(2) or something like that? I'll have to dig it up but it's kinda crazy stuff, mostly around a series construction between square and sine waves using tan^2

 

[Jarhyn]

Let me pose a question similar to yours, but perhaps more concrete. I do NOT know the answer to either this question or yours. :-(

For brevity let
(a,b,c,d,...) denote the remainder after removing portion a, then portion b of what's left (i.e. b*(1-a), then portion c of what's then left and so on.
We wonder whether that process applied to an initial unit will eventually result in Zero (asymptotically). Here are some examples:

• (1/3, 1/3, 1/3, 1/3, ... ) --> Zero // this is the Cantor set
• (1/1000, 1/1000, 1/1000, 1/1000, ...) --> Zero // Although converging slowly it can have no non-zero asymptote)
• (1/2, 1/3, 1/4, 1/5, 1/6, ...) --> Zero // Somewhat harder to prove
• (1/4, 1/4², 1/4³, 1/4⁴, ...) --> Non-zero // (I think) because it empties slower than the SVC set.

But by now, I am lost. I'm not even sure of the final line.

What is sought here, and perhaps also by Jarhyn IIUC, is a process in this form which is "critical", i.e. which fails to converge to Zero but ALMOST succeeds.

I don't have an answer.

BUT yeah, I know I've observed series which converge on numbers like 1/sqrt(2) or something like that? I'll have to dig it up but it's kinda crazy stuff, mostly around a series construction between square and sine waves using tan^2

Later on tonight, I'll put together a Desmos link looking at growth rates and convergence and maybe graphing that and seeing if there's a clear function that pops out. If there's not an answer, well, it shouldn't be too hard to find if we're already talking "in the ballpark" of where answers probably would exist.

Honestly, it's always more fun when the answer isn't known by US, because even when there aren't stakes, it's a neat exercise.

 

[SLD]

So another crazy question, can you prove whether the point 1/4 is in the set or out?

 

[Jarhyn]

So another crazy question, can you prove whether the point 1/4 is in the set or out?

Yes, as you can see if a number is on any of the "edges" of a region that exists, that "book end" will never be eaten. I'm not going to do the work to determine if 1/4 will ever be on a book end, or where book ends are, but you could fairly easily determine if some given number is a "book end" or inside a known gap.

I also wouldn't be helpful in finding a general form, there.

 

[lpetrich]

This is a fractal shape, a shape with a fractional number of dimensions. The fractal or Hausdorff dimension number is calculated from

N = L^D

or

\( \displaystyle{ D = \frac{ \log N }{ \log L } } \)

where D is the fractal dimension number, L is the scaling factor, and N the number of repeats. This formula gives the right result for ordinary numbers of dimensions, and intermediate values for fractal shapes.  List of fractals by Hausdorff dimension

For Cantor dust, it gives log(2)/log(3) = 0.6309

Divide in 3 and remove the center.

Going to two dimensions, we find the Sierpinski carpet or Sierpinski gasket, with dimension log(8)/log(3) = 1.8928.

Divide in 3*3 and remove the center.

In three dimensions, we find the Menger sponge, with dimension log(20)/log(3) = 2.7268.

Divide in 3*3*3 and remove the center and the face centers. That is somewhat different from the previous two, though if one removes only the center, one gets dimension log(26)/log(3) = 2.9656.

There are plenty of other fractal shapes, like the Koch snowflake, with dimension log(4)/log(3) = 1.2619.

There are also space-filling curves, like the Hilbert curve, the Peano curve, the Moore curve. These ones have fractal dimension 2, the dimension of the space that contains them. The Hilbert curve can be generalized to higher numbers of dimensions of containing space.

 

[Swammerdami]

So another crazy question, can you prove whether the point 1/4 is in the set or out?

In arithmetical terms, the Cantor set consists of all real numbers of the unit interval [ 0 , 1 ] that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction.

In base 3, 1/4 = 0.020202020202020...₃

(Admittedly, quoting Wikipedia is not the same as a proof!)

 

[lpetrich]

That result is provable:  Euler's theorem \( a^{\phi(n)} = 1 \mod n \) if a and n are coprime (relatively prime) positive integers and \( \phi(n) \) is Euler's totient function of n, the number of positive integers coprime to n:

\( \displaystyle{ n = \prod_p p^m(p) ,\ \phi(n) = \prod_p p^{m(p)-1} (p-1) } \)

for distinct primes p each to power m(p). For base B and denominator D, \( B^{\phi(D)} - 1 = K D \) and fraction N/D is

\( \displaystyle{ \frac{N}{D} = \frac{K N}{K D} = \frac{K N}{B^{\phi(D)} - 1} } \)

and the final term can easily be expanded as a geometric series in the reciprocal of B, thus giving an infinitely repeating set of digits.

For this case, \(\phi(4) = 2\) and \(3^{\phi{4}} - 1 = 8 = 2 \cdot 4\) and

\( \displaystyle{ \frac{1}{4} = \frac{2}{2 \cdot 4} = \frac{2}{3^2 - 1} = 0.0202020202 \dots } \)

 

[lpetrich]

Oops, number n factorizes into distinct primes p with powers m(p): \( \displaystyle{ n = \prod_p p^{m(p)} } \)