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**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.

Results on the construction of independent sequences and the equivalence of spectral measures complete this paper.

Reviewer: P.Hellekalek