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    Airey, Dylan and Mance, Bill 2015. Unexpected distribution phenomenon resulting from Cantor series expansions. Advances in Mathematics, Vol. 279, p. 372.


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  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Volume 49, Issue 2
  • October 1990, pp. 264-272

Notes on uniform distribution modulo one

  • G. Myerson (a1) and A. D. Pollington (a2)
  • DOI: http://dx.doi.org/10.1017/S1446788700030548
  • Published online: 01 April 2009
Abstract
Abstract

We exhibit a sequence (un) which is not uniformly distributed modulo one even though for each fixed integer k ≥ 2 the sequence (kun) 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 (un) which is u.d. (mod 1) even though no subsequence of the form (ukn + j) is u.d. (mod 1) for any k ≥ 2. We prove that, if the subsequences (ukn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (un) is u.d. (mod I) if the sum of the reciprocals of the primes in P diverges. We show that this result is the best possible of its type.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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