Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-28T21:38:32.937Z Has data issue: false hasContentIssue false

Notes on uniform distribution modulo one

Published online by Cambridge University Press:  09 April 2009

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1990

References

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