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My question is basically,

Let $A_r$ be the set of numbers $x \in [0,1]$ s.t. the inequality $|x-\sqrt{\frac{p}{q}}| < \frac{1}{q^r}$, where $p,q \in \mathbb{N}$, can be satisfied for infinitely many $q$. Then prove $m(A_r)=0$ for $r>2$, and that the Hausdorff dimension of $A_r$ is at most $2/r$. How do I go about proving this?

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Is this a homework? –  Thomas E. Apr 26 '12 at 6:51
    
Instead of asking here, you could look up "Jarnik's Theorem" in a textbook. –  GEdgar Apr 30 '12 at 2:38
    
Ah, is there any online resource available? I tried googling but the results were rather convoluted and confusing –  Cardflow Apr 30 '12 at 19:51

1 Answer 1

Hint: For a given positive integer $q$ those $x \in [0,1]$ such that $|x - \sqrt{p/q}| < 1/q^r$ for some $p$ are in the union of a certain number of intervals, each of a certain length. Can you bound the sum of the lengths of these intervals? What about the sum of the $d$'th powers of the lengths?

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