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 $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally ordered) set of primes. We show that $(f(n))_{n \in \mathbb{N}}$ is uniformly distributed mod 1 if and only if $(f(p))_{p \in \mathcal{P}}$ 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].