On a universal extraction procedure
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 \textit{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 \(\lim_{n\to \infty} (\theta (n+1)- \theta (n))= \infty\).
scientific article; zbMATH DE number 858573