Weakening the hypotheses in the Duffin-Schaeffer conjecture?

Question

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 $C \geq 1$?

Answer 1

To my knowledge here is the best known sufficient condition:

Let $c$ be any positive constant and let $g: [0,\infty)\rightarrow [0,\infty)$ be defined by $g(0)=0$, $$g(x)=x \exp\left(-c (\log (- \log x))(\log \log (-\log x)) \right) \quad \text{ if } ~ 0 < x < 1,$$ and $g(x)=1$ if $x\ge 1$. If $$\sum_{q=1}^{\infty} g\left(\frac{f(q)}{q}\right) \varphi(q)=\infty$$ then $m(\mathbb{R}\setminus E_f)=0$. In particular this implies that the Duffin-Schaeffer Conjecture is true for any $f$ which satisfies $$\sum_{q=16}^{\infty} \frac{\varphi(q) f(q)}{q \exp(c(\log \log q)(\log \log \log q))}=\infty$$ for some $c>0$. Even more particularly, since $(\log \log q)(\log \log \log q)\ll \log q$ this means that if $$\sum_{q=1}^\infty\frac{f(q)}{q^\epsilon}$$ diverges for some $\epsilon >0$ then the Duffin-Schaffer Conjecture holds. However if you replace this sum with the second sum that you asked about (i.e. the one with the power of a logarithm in the denominator), then the answer is not known.

By the way the $1+\epsilon$ result that you mentioned is actually a corollary of a theorem that was originally proved by Glyn Harman (it follows from Theorem 3.7(iii) in his book Metric Number Theory). However the result in the Haynes, Pollington, Velani paper is actually a little stronger.