I have the following set. I need to calculate a lower bound on its measure in order to prove something about some distribution.
$I \subseteq [0,1)$. Take the set $\mathcal{J}(I) = I \bigcap \underset{\underset{(p,q)=1}{Q/2\leq q\leq Q} }\bigcup B(\frac{p}{q},\frac{1}{4q^{2}})$.
I wish to prove that there is a constant $C>0$ s.t for every interval $I\subseteq [0,1)$ there is some $Q\geq Q_{0}$ where $C\lambda(I) Q^{2}\leq \lambda(\mathcal{J}(I))$.
The question stems from a course I take on Diophantine approximations, so maybe tools like Dirichlet approximation theorem can also be of some assistance.