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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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closed as too localized by Gerald Edgar, Yemon Choi, Andrés E. Caicedo, Ryan Budney, Andy Putman Dec 17 '11 at 20:18

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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 Janbaz Dec 15 '11 at 20:41
up vote 3 down vote accepted

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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Many Thanks Dear AH, your answer is sufficient. – Shahrooz Janbaz Dec 15 '11 at 20:43

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