The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: http://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture

My question concerns a special case of the conjecture. It is an immediate corollary to the Borel-Cantelli Lemma that if there exists a function $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that the $E_q$'s are independent (where $\displaystyle E_q = \left[0, \frac{f(q)}{q}\right] \cup \left[1 - \frac{f(q)}{q}, 1\right] \cup \bigcup_{\substack{1 \leq p \leq q \\ \gcd(p,q) = 1}} \left[\frac{p}{q} - \frac{f(q)}{q}, \frac{p}{q} + \frac{f(q)}{q}\right]$. interpreted as the event that $\alpha$ is in the right hand side), then the Duffin-Schaeffer conjecture is true; namely that $\displaystyle \sum_{q=1}^\infty \frac{f(q)}{q} \phi(q) = \infty$ implies that $\mu(\limsup_{q \rightarrow \infty} E_q) = 1$.

So my questions are: 1) Are there any known examples of $f$ such that the $E_q$'s are independent, and 2) is there any hope that a notion of 'almost independent' or 'weakly correlated' can be applied to solve the conjecture? An example of something like this would be Janson's inequality generalizing Chernoff's inequality.