Smith-Volterra-Cantor set

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.

Construction

Similar 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

:$left\; [0,\; frac\{3\}\{8\}\; ight]\; cup\; left\; [frac\{5\}\{8\},\; 1\; ight]$.

The following steps consist of removing subintervals of width $1/2^\{2n\}$ 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

:$left\; [0,\; frac\{5\}\{32\}\; ight]\; cup\; left\; [frac\{7\}\{32\},\; frac\{3\}\{8\}\; ight]\; cup\; left\; [frac\{5\}\{8\},\; frac\{25\}\{32\}\; ight]\; cup\; left\; [frac\{27\}\{32\},\; 1\; ight]$.

Continuing indefinitely with this removal, the Smith-Volterra-Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process:

Properties

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

:$sum\_\{n=0\}^\{infty\}\; 2^n(1/2^\{2n\; +\; 2\})\; =\; frac\{1\}\{4\}\; +\; frac\{1\}\{8\}\; +\; frac\{1\}\{32\}\; +\; cdots\; =\; frac\{1\}\{2\}\; ,$

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

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 less than the measure of the initial interval.

See also

* The SVC is used in the construction of Volterra's function (see external link).

External links

* [http://www.macalester.edu/~bressoud/talks/Volterra-4.pdf "Wrestling with the Fundamental Theorem of Calculus: Volterra's function] , talk by David Marius Bressoud