  Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$
with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some
constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence
of natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish a
quantitative inhomogeneous Khintchine-type theorem in which (i) the points of
interest are restricted to $F$ and (ii) the denominators of the `shifted'
rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a
natural strengthening of the fact that the sequence $(q_nx {\rm \ mod \, }
1)_{n\in \mathbb{N}} $ is uniformly distributed for $\mu$ almost all $x \in F$.
Beyond lacunary, our main theorem implies the analogous quantitative result for
sequences $\mathcal{A}$ for which the prime divisors are restricted to a finite
set of $k$ primes and $A > 2k$.
