Uniform distribution of subpolynomial functions along primes and applications

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 )) n ∈ℕ is uniformly distributed mod1 if and only if ( f ( p )) p ∈P 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].