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.



Special case of Duffin-Schaeffer conjecture
First of all what Ori said about pairwise independence being sufficient for the divergence part of the Borel-Cantelli Lemma is correct- I believe this observation is originally due to Erdos and Renyi. Now I will try to answer both of your questions, in reverse order:


To prove the Duffin-Schaeffer Conjecture it is enough to show that the sets $E\_q$ are *quasi-independent on average*, whenever the sum of their probabilities diverges. Quasi-independence on average is the condition that
$$\sum\_{1\le q,r\le Q} P(E\_q\cap E\_r)\ll \left(\sum\_{1\le q\le Q}P(E\_q)\right)^2,$$
where $\ll$ is Vinogradov notation. Actually it is sufficient just to show that this inequality holds over a subsequence of $Q$ tending to $\infty$. Then by a standard variance argument (for example see Lemma 2.3 of Glyn Harman's book Metric Number Theory) you get that
$$P(\limsup E\_q)>0.$$
Finally there is a zero-one law due to Gallagher which then allows you to conclude that the limsup set has measure $1$.


For the first question, I don't have an example where the sets $E\_q$ are actually bona-fide independent, or even pairwise independent. However there is a trivial estimate for the measure of the intersection of $E\_q$ and $E\_r$ (for $q\not= r$), which is that
$$P(E\_q\cap E\_r)\le 8f(q)f(r).$$
Now let $f:\mathbb{N}\rightarrow [0,\infty)$ be the function defined by $f(q)=1/2q$ if $q$ is prime, and $f(q)=0$ otherwise. Then it is easy to check that when $q$ is prime,
$$P(E\_q)=\frac{\varphi(q)}{q^2}=\frac{1}{q}\left(1-\frac{1}{q}\right)\ge f(q)/2.$$
Therefore in this case we have that the sum of the measures of the sets $E\_q$ diverges, and that all of the sets are pairwise quasi-independent (i.e. not just quasi-independent on average).


