Smith-Volterra-Cantor set

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After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.

In mathematics, the Smith-Volterra-Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowhere dense (in particular it contains no intervals), yet has positive measure.

1 Construction
2 Properties
3 Other fat Cantor sets
4 See also
5 External links

[edit] Construction

Similarly to the construction of the Cantor set, the Smith-Volterra-Cantor set is constructed by removing certain intervals from the unit interval [0, 1].

The process begins by removing the middle 1/4 from the interval [0, 1] (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is

$[0, 3/8] \cup [5/8, 1]$ .

The following steps consist of removing subintervals of width $1 / 2 2 n$ from the middle of each of the $2 n - 1$ remaining intervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving

$[0, 5/32] \cup [7/32, 3/8] \cup [5/8, 25/32] \cup [27/32, 1]$ .

Continuing indefinitely with this removal, the Smith-Volterra-Cantor set is then the set of points that are never removed.

[edit] Properties

By construction, the Smith-Volterra-Cantor set contains no intervals. During the process, intervals of total length

$\sum_{n=0}^{\infty} 2^n(1/2^{2n + 2}) = 1/4 + 1/8 + 1/16 + \cdots = 1/2 \,$

are removed from [0, 1], showing that the set of the remaining points has a positive measure of 1/2.

[edit] Other fat Cantor sets

In general, you can remove r_n from each remaining subinterval at the n-th step of the algorithm, and end up with a Cantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is finite.