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Real numbers having ultimately periodic representations in abstract numeration systems. (English) Zbl 1055.11005

Having previously defined abstract numeration systems associated with infinite ordered regular languages [P. B. A. Lecomte and M. Rigo, Theory Comput. Syst. 34, No. 1, 27–44 (2001; Zbl 0969.68095)] and having extended them for representing real numbers [same authors, Theory Comput. Syst. 35, No. 1, 13–38 (2002; Zbl 0993.68050)], the authors study real numbers whose abstract representation is ultimately periodic. They give syntactic properties of the corresponding words. Furthermore, they show tight links between classical \(\Theta\)-expansions and abstract \(L\)-numerations in the case where \(L\) is the language of all representations of integers in a linear Bertrand numeration system associated with a Pisot number \(\Theta\).

MSC:

11A63 Radix representation; digital problems
68Q45 Formal languages and automata
68R15 Combinatorics on words
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