Thursday, September 26, 2013

Magnifying sets of real numbers

If S is a set of real numbers and a is a real number, let aS={ax:xS} be the set you get by magnifying S by a factor a.

Here's a funny thing. Some sets get bigger when magnified and some get smaller. For instance, if we take the interval [0,1] and magnify it by a factor of two, we get the interval [0,2], which is intuitively "twice as big". But if we take the natural numbers N and magnify them by a factor of two, 2N will be the even natural numbers, and so 2N will intuitively be "twice as small" as N.

Next observe that if R−[0,1] is the real numbers outside the interval [0,1], then 2(R−[0,1])=R−[0,2] is smaller. Magnifying a set by a factor greater than 1 magnifies both the filled in parts of the set and the holes in the set. The effect of this on the intuitive "size" of the set will depend on the interaction between the holes and the filled in parts.

And if we take the Cantor set C, then magnifying it by a factor of three makes the set be intuitively twice as large. I.e., 3C=C∪(2+C). This makes it very intuitive that the dimension of the set is log 2 / log 3 (which is indeed its Hausdorff dimension). For intuitively if we have an n-dimensional set, and we magnify it by a factor of a, its size is an. So if n is the dimension of the Cantor set, then 2=3n, and so n is log 2 / log 3.

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