Uniform distribution modulo one on subsequences

Abstract:

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 $p\in \mathcal {P}$.