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 $A > 2k$.

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具有受限分母的 M0 集上的非齐次丢番图近似

令 $F \subseteq [0,1]$ 是一个支持概率测度 $\mu$ 的集合，其性质为 $|\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ 对于一些常数 $ A > 0 $。令 $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ 是一个自然数序列。如果 $\mathcal{A}$ 是空缺的并且 $A >2$，我们建立了一个定量的非齐次 Khintchine 型定理，其中 (i) 兴趣点仅限于 $F$ 和 (ii) 的分母移位的有理数仅限于 $\mathcal{A}$。该定理可以看作是序列 $(q_nx {\rm \ mod \, } 1)_{n\in \mathbb{N}} $ 均匀分布于 $\mu$ 几乎所有$x \in F$。超越空缺，