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Notes on uniform distribution modulo one

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

G. Myerson  and
A. D. Pollington
G. Myerson
Affiliation:
School of Mathematics, Physics, Computing and Electronics, Macquarie UniversityAustralia2109
A. D. Pollington
Affiliation:
Department of MathematicsBrigham Young UniversityProvo, Utah 84602, U.S.A.
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Abstract

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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.

MSC classification

Secondary: 11K06: General theory of distribution modulo $1$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 2 , October 1990 , pp. 264 - 272
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, (Wiley, 1974).Google Scholar