Inhomogeneous Diophantine Approximation onM0-Sets with Restricted Denominators

Let F ⊆ [0,1] be a set that supports a probability measure μ with the property that |μ (t)| (log |t|)-A for some constant $ A> 0 $. Let A= (qₙ)_n∈N be a sequence of natural numbers. If 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 A. The theorem can be viewed as a natural strengthening of the fact that the sequence (qₙx mod 1)_n∈N is uniformly distributed for μ almost all x ∈ F. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences 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.