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?