Independent numeration systems and syndeticity
The famous Cobham's theorem [\textit{A.~Cobham}, Math.~System Theory 3, 186--192 (1969; Zbl 0179.02501)] asserts that a set \(X\) of natural numbers is recognizable in two multiplicatively independent bases if and only if \(X\) is ultimately periodic. In the original paper the bases are natural numbers. Similar statements have been proved for numeration systems as defined by \textit{A. Bertrand-Mathis} [Acta Math. Hung. 54, No. 3--4, 237--241 (1989; Zbl 0695.10005)] and for subsets of \(\mathbb N ^k\). All these generalizations needed the hypothesis the base of the numeration system is a Pisot number. The main result of the paper under review asserts the syndeticity of any infinite set of natural numbers which is \(\theta\)-recognizable and \(\theta '\)-recognizable for some multiplicatively independent real numbers \(\theta, \theta ' >1\). As an application one derives a result of Cobham's theorem type for subsets of \(\mathbb N ^k\). This result has been obtained independently by a different method by \textit{A.~Bès} [J.~Symb. Logic 65, No. 1, 201--211 (2000; Zbl 0958.03025)]. (Reviewer's remark: the reference [6] in the paper under review should be \textit{A.~Cobham} [Math.~System Theory 3, 186--192 (1969)]).
scientific article; zbMATH DE number 1238867