## Some metric properties of subsequences.(English)Zbl 0716.11038

Let X be a compact metrizable space and let $${\mathcal U}$$ be a finite family of X-valued sequences. The set of these sequences will be denoted be $$X^{\infty}$$. $${\mathcal U}$$ is called statistically independent if $\lim_{N\to \infty}((1/N)\sum_{n<N}\prod_{u\in {\mathcal U}}f(u_ n)- \prod_{u\in {\mathcal U}}(1/N)\sum_{n<N}f(u_ n))=0$ for any f : $$X\to {\mathbb{C}}$$ continuous. If $${\mathcal F}$$ is a family of sequences $$\sigma\subset {\mathbb{N}}$$ with $$\lim_{n\to \infty} \sigma_ n=\infty$$, then an element u of $$X^{\infty}$$ is called $${\mathcal F}$$-independent, if the family $$\{$$ $$u\circ \sigma:\sigma \in {\mathcal F}\}$$ is statistically independent. u is said to be ($${\mathcal F},\mu)$$-independently distributed if each sequence $$u\circ \sigma$$ ($$\sigma\in {\mathcal F})$$ is $$\mu$$- uniformly distributed and if u is $${\mathcal F}$$-independent. A sequence $$\sigma$$ of positive integers is called regular if there is a subset A of $${\mathbb{N}}$$ with asymptotic density 1 such that $$\sigma$$ restricted to A is one-to-one. Finally, a family $${\mathcal F}$$ of sequences of positive integers is called sparse [“scattered” is also in use, cf. J. Coquet and the author, Compos. Math. 51, 215-236 (1984; Zbl 0537.10030)] if, for all $$\sigma$$,$$\tau$$ in $${\mathcal F}$$, $$\sigma\neq \tau$$ implies $\lim_{N\to \infty}(1/N) card\{n<N :\;\sigma_ n\neq \tau_ n\}=0.$ The author proves: Theorem 1. If $${\mathcal F}$$ is a countable sparse family of regular sequences and if $$\mu$$ is a Borel probability measure on X, then (with respect to product measure) almost all sequences u in $$X^{\infty}$$ are $${\mathcal F}$$-independent, each sequence $$u\circ \sigma$$ ($$\sigma\in {\mathcal F})$$ being $$\mu$$-uniformly distributed. In Theorem 2, the existence of $${\mathcal F}$$-independent u such that $$\{$$ $$u\circ \sigma \}$$ is equi-$$\mu$$-uniformly distributed is shown for certain countable families $${\mathcal F}$$ of strictly increasing sequences of positive integers. Theorem 3. For almost all real $$\theta >1$$, the sequence $$(\theta^ n)_{n\geq 0}$$ is $${\mathcal F}$$-independent for $${\mathcal F}=\{p\in {\mathbb{R}}[x]:p({\mathbb{N}})\subset {\mathbb{N}}\}$$, each sequence $$(\theta^{p(n)})_{n\geq 0}$$ being uniformly distributed modulo 1 (p$$\in {\mathcal F}).$$
Results on the construction of independent sequences and the equivalence of spectral measures complete this paper.
Reviewer: P.Hellekalek

### MSC:

 11K41 Continuous, $$p$$-adic and abstract analogues 11K31 Special sequences

Zbl 0537.10030
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