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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.
Reviewer: P.Hellekalek

MSC:

11K41 Continuous, \(p\)-adic and abstract analogues
11K31 Special sequences

Citations:

Zbl 0537.10030
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