Uniform distribution of subpolynomial functions along primes and applications

Abstract

Let H be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let f H be a subpolynomial function. Let P be the sequence of naturally ordered primes. We show that (f(n))(nN) is uniformly distributed mod1 if and only if (f (p))(pP) is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].