Combinatorics, Automata and Number Theory

numeration systems The motivation for the introduction of abstract numeration systems stemsfrom the celebrated theorem of Cobham dating back to 1969 about the so-called recognisable sets of integers in any integer base numeration system.An abstract numeration system is simply an infinite genealogically ordered(regular) language. In particular, this notion extends the usual integer basenumeration systems as well as more elaborated numeration systems such asthose based on a Pisot number. In this general setting, we study in detailsrecognisable sets of integers, i.e., the corresponding representations are ac-cepted by a finite automaton. The main theme is the link existing betweenthe arithmetic properties of integers and the syntactical properties of thecorresponding representations in a given numeration system. Relationshipwith automatic sequences and substitutive words is also investigated, pro-viding an analogue to another famous result of Cobham from 1972 aboutk-automatic sequences. Finally, the chapter ends with the representationof real numbers in an abstract numeration system. Chapter 4 by J. Cassaigne and F. Nicolas