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Uniform distribution modulo one on subsequences

Authors:	Chris Hill

Institution:	Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801

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 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 $p\in \mathcal{P}$ .

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