  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].
