For any function $f:\mathbb{N}\to \mathbb{N}$ the property that, for almost all $\alpha$$$\left|\alpha -\frac{p}{q}\right|<\frac{f(q)}{q}$$has infinitely many solutions with $(p,q)=1$, is equivalent to$$\sum _{q\ge 1}\varphi (q)\frac{f(q)}{q}=\infty .$$