## 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:

 [1] Zygmund, Trigonometric Series I, II (1979) · Zbl 0628.42001 [2] Lyons, Studia Math. 86 pp 59– (1987) [3] DOI: 10.1007/BF01475864 · doi:10.1007/BF01475864 [4] DOI: 10.2307/1971372 · Zbl 0583.43006 · doi:10.2307/1971372 [5] DOI: 10.2307/2044653 · Zbl 0545.28007 · doi:10.2307/2044653 [6] DOI: 10.2307/2044398 · Zbl 0468.28013 · doi:10.2307/2044398 [7] Katznelson, An Introduction to Harmonic Analysis (1976) [8] Graham, Essays in Commutative Harmonic Analysis (1979) · Zbl 0439.43001 · doi:10.1007/978-1-4612-9976-9 [9] Cornfeld, Ergodic Theory (1982) · doi:10.1007/978-1-4615-6927-5 [10] DOI: 10.1007/BF02756822 · Zbl 0283.60053 · doi:10.1007/BF02756822 [11] Bari, A Treatise on Trigonometric Series I, II (1964) [12] Smorodinsky, Ergodic Theory Entropy (1971) · doi:10.1007/BFb0066086 [13] none, Amer. Math. Soc. Transl. 39 pp 1– (1964) · Zbl 0154.15703 · doi:10.1090/trans2/039/01 [14] Rokhlin, Izv. Akad. Nauk SSSR Ser. Mat. 25 pp 499– (1961) [15] none, Moscov. Gos. Univ. U?. Zap. 165 pp 79– (1954) [16] Pjatecki?-Šapiro, Moscov. Gos. Univ. U?. Zap. 155 pp 54– (1952) [17] Petit, C. R. Acad. Sc. 280 pp 17– (1975) [18] DOI: 10.1090/S0002-9904-1952-09580-X · Zbl 0046.11504 · doi:10.1090/S0002-9904-1952-09580-X [19] none, Gesammelte Abhandlungen 1 pp 563– (1968)
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