Inhomogeneous Diophantine Approximation on M₀-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 = (q_n)_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 (q_nxmod1)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.