For any non-increasing f with f(q) -> 0 and sum f(q) = infinity, there is a set of denominators S with sum_{q in S} f(q) = infinity while the set of approximated numbers has Lebesgue measure zero.
Hauke, Quantitative inhomogeneous Diophantine approximation fo r systems of linear forms, Proc
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General Duffin--Schaeffer-type counterexamples in diophantine approximation
For any non-increasing f with f(q) -> 0 and sum f(q) = infinity, there is a set of denominators S with sum_{q in S} f(q) = infinity while the set of approximated numbers has Lebesgue measure zero.