Inhomogeneous Diophantine Approximation on M0-Sets with Restricted Denominators

Abstract

Abstract 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 (1) the points of interest are restricted to $F$ and (2) 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 \textrm{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 $A> 2k$. Loosely speaking, for such sequences, our result can be viewed as a quantitative refinement of the fundamental theorem of Davenport, Erdös, and LeVeque (1963) in the theory of uniform distribution.