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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 someconstant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequenceof natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish aquantitative inhomogeneous Khintchine-type theorem in which (i) the points ofinterest are restricted to $F$ and (ii) the denominators of the `shifted'rationals are restricted to $\mathcal{A}$. The theorem can be viewed as anatural 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 forsequences $\mathcal{A}$ for which the prime divisors are restricted to a finiteset of $k$ primes and $A > 2k$.

Inhomogeneous Diophantine Approximation on $M_0$-sets with restricted denominators