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On uniformly distributed sequences of integers and Poincaré recurrence III
Published online by Cambridge University Press: 17 April 2009
Abstract
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Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence 
contained in S is Hartman uniform distributed on G if
for any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed then
In this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 68 , Issue 2 , October 2003 , pp. 345 - 350
- Copyright
- Copyright © Australian Mathematical Society 2003
References
[1]Nair, R., ‘On uniformly distributed sequences of integers and Poincaré recurrence’, Indag. Math. (N.S.) 9 (1998), 55–63.Google Scholar
[2]Nair, R., ‘On Hartman uniform distribution and measures on compact groups’, in Harmonic Analysis and Hypergroups, (Anderson, M., Singh, A.I. and Ross, K., Editors), Trends in Mathematics 3 (Birkhauser, Boston, M.A., 1998), pp. 59–75.CrossRefGoogle Scholar
[3]Nair, R., ‘On uniformly distributed sequences of integers and Poincaré recurrence II’, Indag. Math. (N. S.) 9 (1998), 405–415.Google Scholar
[4]Nair, R., and Weber, M., ‘On random perturbations of some intersective sets’, Indag. Math. (to appear).Google Scholar
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