Distribution modulo 1 of some oscillating sequences. III

For some oscillating functions, such as h(x) = x(pi) log 3 x cos x, we we consider the distribution properties modulo I (density, uniform distribution) of the sequence h(n), n greater than or equal to 1. We obtain positive and negative results covering the case when the factor x(pi) log(3) X is replaced by an arbitrary function f of at most polynomial growth belonging to any Hardy field. (The latter condition may be viewed as a regularity growth condition on f.) Similar results are obtained for the subsequence h(p), taken over the primes p = 2, 3, 5,....