In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has infinitely many integer solutions p,q for almost all $\alpha$. (Often you also assume that $p$ and $q$ must be coprime, but this is not the point in my question). A problem of this type is for example the famous (unsolved) Duffin-Schaeffer conjecture, see http://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture.
The problem can also be written in the following form: Let $A_1, A_2, \dots$ be intervals on the torus (of length $\leq 1$), which are symmetric around 0. Let $\psi_1, \psi_2, \dots$ denote the Lebesgue measure (that is, the length) of these intervals. Under which conditions on $\psi_1, \psi_2, \dots$ do we have $$ \sum_{n=1}^\infty \mathbf{1}_{A_n} (n \alpha) = \infty $$ for almost all $\alpha$ (here $\mathbf{1}_A$ is the indicator function of $A$. Also, everything is understood to take place on the torus, so the indicators are extended with period 1.)
Now you may generalize the situation in a first step to the case when the intervals $A_1, A_2, \dots$ are not necessarily symmetric around $0$. Then you get a problem in inhomogeneous Diophantine approximation. This type of question is also quite well-investigated.
Now my question is the following: what happens is if don't assume that $A_1, A_2, \dots$ are intervals, but if they may denote any measurable sets in $[0,1]$. Again, write $\psi_1, \psi_2, \dots$ for the measure of these sets. Under which conditions on $\psi_1, \psi_2, \dots$ do we have $$ \sum_{n=1}^\infty \mathbf{1}_{A_n} (n \alpha) = \infty $$ for almost all $\alpha$? Are there any sufficient conditions? (Note that a necessary condition is the divergence of the sum of the measures, by Borel-Cantelli). Does it help if the sequence $(\psi_n)_{n \geq 1}$ is monotonic? Are there any known results at all about this general problem?
(Remark: This question is distantly related to Khintchin's conjecture, which was disproved by Marstrand in 1970. See http://plms.oxfordjournals.org/content/s3-21/3/540.full.pdf.)
(Remark 2: The sequence $(n \alpha)_{n \geq 1}$ mod 1 has an interpretation in terms of ergodic theory, but I don't think that this will help here.)
