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Abstract β -expansions and ultimately periodic representations

Michel Rigo, Wolfgang Steiner (2005)

Journal de Théorie des Nombres de Bordeaux

For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is ( β ) if the dominating eigenvalue β > 1 of the automaton accepting the language is a Pisot number. Moreover, if β is neither a Pisot nor a Salem number, then there exist points in ( β ) which do not have any ultimately periodic representation.

Addendum to: On volumes of arithmetic quotients of SO (1, n)

Mikhail Belolipetsky (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.

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