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?



existence of fractal
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 $2/\tau$.


