Inhomogeneous Diophantine Approximation onM0-Sets with Restricted Denominators

Abstract Let F 0, 1 be a set that supports a probability measure with the property that | (t) | (|t|) ^-A for some constant A 0. Let A= (qₙ) ₍₍ be a sequence of natural numbers. If A is lacunary and A2, 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) ₍₍ is uniformly distributed for 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.