Talk:Smith–Volterra–Cantor set

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Nowhere dense

The same content is present at nowhere dense but I felt this deserves a separate page. I'll be creating a page on Volterra's function in the near future if someone doesn't beat me to it :-) - Gauge 06:07, 23 Aug 2004 (UTC)

There is a slight inconsistency here. Either the intervals removed at each step are the "middle quarter of the remaining intervals" or they are centred on a/2^n. But they cannot be both. The first leads to measure of 0.5, the second to measure of 0.53557368... --Henrygb 17:17, 25 Apr 2005 (UTC)

Thanks for noticing. I noticed that the intervals you gave seemed to be off, so I corrected them (hopefully :-)). - Gauge 01:42, 30 Apr 2005 (UTC)

You are right - somehow I multipled 3 by 2 and got 12. Thanks --Henrygb 22:53, 30 Apr 2005 (UTC)

Can somebody please help me understand?

Let the set be called S. By construction, S contains no intervals (i.e. S contains points that are seperate from each other.). And the measure of a single point is 0. So how can the total measure be 1/2? On the other hand, the total length of removal is 1/2, hence remaining length must be 1/2. Hence, S must contains intervals of length greater than 0. Can somebody please help me resolve this? 108.162.157.141 (talk) 01:53, 28 November 2013 (UTC)[reply]

Intuition must adapt to facts! Yes, it is hard. For now, your intuition tells you that the measure of a set is the sum of lengths of intervals. And your logic already tells you the opposite. Your intuition must adapt. It is a hard internal work. For even harder case, see Weierstrass function. Such is the life. Boris Tsirelson (talk) 06:18, 28 November 2013 (UTC)[reply]

Please, explain the hausdorf dimension

This article needs to explain the hausdorff dimension of the Generalized Cantor set, as listed on the wikipage List of fractals by Hausdorff dimension, which is shown as

$Dimension={\frac {-\log(2)}{\log \left(\displaystyle {\frac {1-\gamma }{2}}\right)}}$

Some refs:

An Exploration of the Cantor Set by Christopher Shaver
Hausdorff measure of p-Cantor sets by C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler
Topological dimensions, Hausdorff dimensions and fractals by Yuval Kohavi and Hadar Davdovich
Dimension of the Cantor set by Michael Damron

Boris Tsirelson (talk) 20:11, 2 January 2017 (UTC)[reply]

Naming

Could someone report on whether the particular name Smith–Volterra–Cantor set is attested in the references, or at least in the wild? The basic underlying idea here (not necessarily the specific sequence using ${\frac {1}{4^{n}}}$ ) is pretty standard and will inevitably come up in almost any real analysis course when clarifying the distinction between measure and category. The specific name used here, on the other hand, I do not recall seeing outside Wikipedia.

I'm concerned that this name could have been what an editor thought the construction should be called, which is something we're not supposed to do, though that may not have been as clear in 2004 when the article was first named. Gauge are you still around to comment? --Trovatore (talk) 19:03, 21 August 2023 (UTC)[reply]

Looking at mathse and mo, there are more posts using the terminology "fat Cantor set". Most of the posts using "Smith-Volterra-Cantor set" actually point to Wikipedia. It could be worth asking the professional mathematics community on mathoverflow about it. PatrickR2 (talk) 00:58, 22 August 2023 (UTC)[reply]

That's about what I'd have expected. Unless one of the references actually uses the term from the title, I think we should move this article to fat Cantor set, and rewrite it a bit. --Trovatore (talk) 01:11, 22 August 2023 (UTC)[reply]

No objection on my part, if we can't find a valid reference for it. (We should keep "Smith-Volterra-Cantor set" at least as a redirect.) But I have not looked at Aliprantis & Burkinshaw, mentioned as a source. PatrickR2 (talk) 17:24, 22 August 2023 (UTC)[reply]

Looking at google books, it seems like Fat Cantor Set occurs in some trustworthy books (like Frank Jones' Lebesgue Integration on Euclidean Space) while others like Royden and Carothers use Generalized Cantor set.

The Smith-Volterra-Cantor set is a specific example of a fat cantor set or generalized cantor set, so I support moving the article name to either one of those options and citing the Smith-Volterra-Cantor set as an early example. Some nice history on that specific set and the contributions of the different names can be found here. If I had to vote I'd go with 'generalized cantor set' as some pretty heavy-duty authors use it but would support a move to Fat Cantor Set as well. Brirush (talk) 22:44, 22 August 2023 (UTC)[reply]

@Brirush: do you know of any independent attestation of the specific name Smith–Volterra–Cantor set? Even with the changes you outline, I'm still concerned about using this name if it's not the standard name "in the wild" for the set described. If it was invented for the Wikipedia article, well, that's not supposed to happen. That said, it was 19 years ago and WP is very influential, so it's possible that the name has propagated back into the community, and in that case I suppose we can use it. --Trovatore (talk) 23:12, 22 August 2023 (UTC)[reply]

Using Google books, 'Smith-Volterra-Cantor' does not show any results before 2003. Well, it does, giving about 10 books, but searching within the books they discuss Volterra and Smith separately. I was unable to find the specific use of that name. It appears frequently after 2004 or so, which is around when the article was created, so I'm sure that Wikipedia popularized the name. However, the set was associated with Smith and Volterra before, it just (as far as I'm able to discern) didn't have that specific name before. It has since then been attested in several reliable sources.Brirush (talk) 23:38, 22 August 2023 (UTC)[reply]

Thanks, Brirush. We should try hard to keep this kind of thing from happening. But if it's already happened, it's not really our place to undo it. I still do think we should move to "fat Cantor set" or "generalized Cantor set" and reprioritize the article so as not to be about the specific

{\frac {1}{4^{n}}}

construction. --Trovatore (talk) 00:05, 23 August 2023 (UTC)[reply]

I agree with the move and reprioritization. Brirush (talk) 00:49, 23 August 2023 (UTC)[reply]

Agree also. PatrickR2 (talk) 17:16, 23 August 2023 (UTC)[reply]

I agree. I've only ever heard this called the fat Cantor set. - CRGreathouse (t | c) 16:28, 25 August 2023 (UTC)[reply]

In the hey-day of chaos theory in the 1980's, when fractals were the only thing anyone ever talked about, this was called the "fat Cantor set". As I recall, there was an especially fine issue of Physics Today (or was it American Mathematical Monthly? Sheesh.) that was devoted to all things fractal and chaotic, and it is where I learned of the fat Cantor set. 67.198.37.16 (talk) 03:08, 28 November 2023 (UTC)[reply]