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

Abstract

Let $F\subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\hat{\mu }(t)|\ll (\log |t|{)}^{-A}$ 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 $(q_{n}x\mathrm{mod}1{)}_{n\in \mathbb{N}}$ 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 $A>2k$.

58 pages