Smith-Volterra-Cantor set

In mathematics, the Smith-Volterra-Cantor set is a set of points on the real line $\mathbb{R}$ satisfying the following interesting combination of properties:

SVC is nowhere dense (in particular it contains no intervals),
SVC has positive measure.

It is a close relative of the Cantor set, 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 second step consists of removing the middle quarter of these two sets (the middle "quarters" now of size 1 / 64), and so on. More technically, this is done at each step by considering dyadic fractions of the form

$\frac{a}{2^n}$

contained in [0,1] where a is an odd number. These fractions are then taken to be midpoints of the remaining subintervals, and

$\frac{1}{2^{2n + 1}}$

is then removed from either side of each of these midpoints (removing a total of $\frac{1}{2^{2n}}$ around each such point).

Clearly as n approaches $\infty$ the number of intervals goes to 0, thus illustrating the third property mentioned above. It can also easily be shown to have a positive measure of 1 / 2 by observing that intervals of total length

$2^0(1/4) + 2^1(1/16) + 2^2(1/64) + \ldots + 2^n(1/2^{2n + 2}) + \ldots = 1/4 + 1/8 + 1/16 + \ldots = 1/2 \,$

are removed from [0,1] during the construction.

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

External link

http://www.macalester.edu/~bressoud/talks/Volterra-4.pdf

Categories: Measure theory | Topology | General topology | Fractals

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