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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?

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    $\begingroup$ Yes, and yes. But this is not the place to ask. See the FAQ for suggestions of other places to ask. $\endgroup$
    – Gerald Edgar
    Commented Dec 15, 2011 at 20:14
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    $\begingroup$ Yes, take a look at the Wikipedia page about Hausdorff dimension. $\endgroup$
    – Guillaume Brunerie
    Commented Dec 15, 2011 at 20:18

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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 $2/\tau$.

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  • $\begingroup$ Many Thanks Dear AH, your answer is sufficient. $\endgroup$
    – Shahrooz
    Commented Dec 15, 2011 at 20:43

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