Smith-Volterra-Cantor set

From Wikipedia, the free encyclopedia

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

Contents

[edit] 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}\right] \cup \left[\frac{5}{8}, 1\right].

The following steps consist of removing subintervals of width 1 / 22n from the middle of each of the 2n − 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}\right] \cup \left[\frac{7}{32}, \frac{3}{8}\right] \cup \left[\frac{5}{8}, \frac{25}{32}\right] \cup \left[\frac{27}{32}, 1\right].

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:

[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}) = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \frac{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 rn 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.

[edit] See also

[edit] External links