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

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 $A > 2k$.