Uniform distribution modulo one on subsequences

Let $\mathcal{P}$ be a set of primes with a divergent series of reciprocals and let $\mathcal{K} = \mathcal{K}(\mathcal{P} )$ denote the set of squarefree integers greater than one that are divisible only by primes in $\mathcal{P}$. G. Myerson and A. D. Pollington proved that $(u_{n})_{n\geq 1}\subset [0,1)$ is uniformly distributed (mod 1) whenever the subsequence $(u_{kn})_{n\geq 1}$ is uniformly distributed (mod 1) for every $k$ in $\mathcal{K}$. We show that in fact $(u_{n})_{n\geq 1}$ is uniformly distributed (mod 1) whenever the subsequence $(u_{pn})_{n\geq 1}$ is uniformly distributed (mod 1) for every $p\in \mathcal{P}$.