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Uniform distribution and ergodicity. (Répartition et ergodicité.) (French) Zbl 0392.10048

Sémin. Delange-Pisot-Poitou, 19e Année 1977/78, Théor. des Nombres, Fasc. 1, Exp. No. 10, 12 p. (1978).
The author studies ergodic properties of transformations \((x,g)\to (Tx,g\cdot f(x))\) on \(X\times G\), \(X\) a probability measure space, \(G\) a compact metric group. The results are related with papers of H. Furstenberg [Am. J. Math. 83, 573–601 (1961; Zbl 0178.38404)] and J. P. Conze [Equirépartition et ergodicité de transformation cylindriques, Univ. de Rennes (1977)]. As a consequence one obtains an extremely surprising phenomena related with an important work of W. A. Veech [Ann. Math. (2) 94, 125–138 (1971; Zbl 0226.43001)].
Some questions of uniform distribution: A sequence of positive integers \(n_j\) is a “uniformly distributed sequence generator” if for any compact group \(G\) and any subset \(\{a_n,\ 1\le n<\infty\}\) generating a dense subgroup the sequence \(z_m = \displaystyle\prod_{j\le m} a_{n_j}\) is uniformly distributed in \(G\).
Veech did not only prove the existence of such generators (the case of well distributed sequence generators is still open, but see V. Losert and the reviewer [Uniform distribution and the mean ergodic theorem, Invent. Math. 50, 65–74 (1978; Zbl 0414.22006)]) but also gave an explicit and natural construction associated with an arbitrary number normal to base \(m\) and an interval \((a, b)\subseteq [0,1)\). It is shown here that this method works if and only if \(m\ne 2\) or \((a,b)\ne (1/6,5/6)\)!
The author proves also that for any completely u.d. sequence \((x_n)\) and any nontrivial interval \(I\), the sequence \((n,\ \sum_{h\le n}\chi_I(x_h))\) is u.d. in \(\mathbb Z^2\) the sense of Hartman. This implies that for any nontrivial interval \(I= (a,b)\):
\(\left(\displaystyle\sum_{h\le n}\chi_I(x_h) - n(b - a)\right)\) is u.d. in \(\mathbb Z\) for almost all sequences \((x_h)\).
[This article was published in the book announced in this Zbl. 383.00003.]

MSC:

11K06 General theory of distribution modulo \(1\)
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
47A35 Ergodic theory of linear operators
28D99 Measure-theoretic ergodic theory
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