Uniform distribution modulo one on subsequences

Abstract

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