## Abstract

We exhibit a sequence (u_{n}) which is not uniformly distributed modulo one even though for each fixed integer k > 2 the sequence (ku_{n}) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (u_{n}) which is u.d. (mod 1) even though no subsequence of the form {u_{kn+j}) is u.d. (mod 1) for any k > 2. We prove that, if the subsequences (u_{kn}) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set p, then (u_{n}) is u.d. (mod 1) if the sum of the reciprocals of the primes in p diverges. We show that this result is the best possible of its type.

Original language | English |
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Pages (from-to) | 264-272 |

Number of pages | 9 |

Journal | Journal of the Australian Mathematical Society |

Volume | 49 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1990 |