# existence of fractal [closed]

I have a question about fractals;

Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$? If yes, do we have any method to construct such fractal?

-
Yes, and yes. But this is not the place to ask. See the FAQ for suggestions of other places to ask. –  Gerald Edgar Dec 15 '11 at 20:14
Yes, take a look at the Wikipedia page about Hausdorff dimension. –  Guillaume Brunerie Dec 15 '11 at 20:18
Thanks, I found it. –  Shahrooz Dec 15 '11 at 20:41
Yes, there are many examples, and constructible ones abound. I'll just mention one example: There is a theorem of Jarnik from the 1920's or 30's that says that for any $\tau\ge 2$ the collection of real numbers $x$ for which the inequality $$|x-a/q|\le 1/q^{\tau}$$ has infinitely many solutions $a,q\in\mathbb{N}$, has Hausdorff dimension $2/\tau$.