On a universal extraction procedure. (Sur un procédé universel d’extraction.)(French)Zbl 0843.11014

Let $$\theta: \mathbb{N}\to \mathbb{N}$$ be a strictly increasing sequence of positive integers, set $$E= \theta (\mathbb{N})$$ and let $$n= \sum_{k\geq 0} \varepsilon_k (n) \cdot d^k$$ be given in $$d$$-ary digital expansion. Following P. Liardet [Acta Arith. 55, 119-135 (1990; Zbl 0716.11038)] the author considers sequences of the type $$\sigma_E (n)= \sum_{k\geq 0} \varepsilon_{\theta (k)} (n) d^k$$. Periodicity and quasi-periodicity properties are investigated. For instance, it is proved that $$u\circ \sigma_E$$ is $$d$$-automatic provided that $$E$$ is ultimately periodic and $$u$$ is $$d$$-automatic. Furthermore $$\sigma_E$$ is quasi-periodic if and only if $$\lim_{n\to \infty} (\theta (n+1)- \theta (n))= \infty$$.
Reviewer: R.F.Tichy (Graz)

MSC:

 11B85 Automata sequences 11A67 Other number representations

Zbl 0716.11038
Full Text:

References:

 [1] Christol, G., Kamae, T., France, M. Mendès, Rauzy, G., Suites algébriques, automates et substitutions, Bull. Soc. Math. France108 (1980), 401-419. · Zbl 0472.10035 [2] Cobham, A., Uniform tag séquences, Mathem. Syst. Theory6, 1972, p.164-192. · Zbl 0253.02029 [3] Gel’fond, A.O., Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta ArithmeticaXIII, 1968, 259-265. · Zbl 0155.09003 [4] Kuipers, L., Niederreiter, H., Uniform distribution of sequences, Pure and applied mathematics, Wiley, New-York-NY, 1974. · Zbl 0281.10001 [5] Liardet, P., Some metric properties of subsequences, Acta ArithmeticaLV, 1990, 119-135. · Zbl 0716.11038
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