The Duffin-Schaeffer conjecture is a conjecture in metric number theory, which asks for a given function $f : \mathbb{R} \rightarrow \mathbb{R}^+$ the measure of the set of real numbers $\alpha$ such that the inequality $$\displaystyle \left | \alpha - \frac{p}{q} \right| < \frac{f(q)}{q}$$ has infinitely many solutions in integers $p,q$ with $\gcd(p,q) = 1$. The conjecture asserts that the set of solutions $\alpha$ has full measure if and only if $\displaystyle \sum_{q=1}^\infty \frac{f(q)}{q} \phi(q)$ diverges, where $\phi(q)$ is the Euler totient function.

Let the set of solutions (which depends on the function $f$) be denoted $E_f$. Then it is known (Haynes, Pollington, Velani http://arxiv.org/abs/0811.1234) that a sufficient condition for $m(\mathbb{R} \setminus E_f) = 0$ is for $\displaystyle \sum_{q=1}^\infty \left(\frac{f(q)}{q}\right)^{1 + \epsilon} \phi(q) = \infty$. My question concerns other possible sufficient conditions. In particular, we know (from Duffin and Schaeffer themselves) that it is not sufficient for $\displaystyle \sum_{q=1}^\infty f(q) = \infty$. Is it sufficient for the sum $\displaystyle \sum_{q=1}^\infty \frac{f(q)}{q^\epsilon}$ to diverge for any $\epsilon > 0$? What about $\displaystyle \sum_{q=1}^\infty \frac{f(q)}{\log^C(q)} = \infty$ for some $C \geq 1$?