Talk:Smith–Volterra–Cantor set

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)

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)

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)

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)