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Mixing and asymptotic distribution modulo 1. (English) Zbl 0645.10042

If \(\mu\) is a probability measure which is invariant and ergodic with respect to the transformation \(x\mapsto qx\) on the circle \(\mathbb{R}/\mathbb{Z}\), then according to the ergodic theorem, \(\{q^nx\}\) has the asymptotic distribution \(\mu\) for \(\mu\)-a.e. \(x\). On the other hand, Weyl showed that when \(\mu\) is Lebesgue measure, \(\lambda\), and \(\{m_j\}\) is an arbitrary sequence of integers increasing strictly to \(\infty\), the asymptotic distribution of \(\{m_jx\}\) is \(\lambda\) for \(\lambda\)-a.e. x.
In this article, the author investigates the asymptotic distributions of \(\{m_ jx\}\) \(\mu\)-a.e. for fairly arbitrary \(\{m_ j\}\) under some strong mixing conditions on \(\mu\). The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to \(\mu\). The ideas extend to the situation of a sequence of transformations \(x\mapsto q_nx\) where invariance is not present. This gives information about many Riesz products and Bernoulli convolutions. Finally, the theory is applied to resolve some questions about \(H\)-sets.
Reviewer: Russell Lyons

MSC:

11K06 General theory of distribution modulo \(1\)
28D05 Measure-preserving transformations
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References:

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