Inhomogeneous Diophantine Approximation on M-0-Sets with Restricted Denominators

Let F subset of [0, 1] be a set that supports a probability measure mu with the property that vertical bar(mu) over cap (t)vertical bar << (log vertical bar t vertical bar)(-A) for some constant A > 0. Let A = (q(n))(n is an element of 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(n)xmod1)(n is an element of N) is uniformly distributed for mu almost all x is an element of 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, Erdos, and LeVeque (1963) in the theory of uniform distribution.