Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems
Daniel Dombek, Zuzana Masáková, Tomáš Vávra · Journal de Théorie des Nombres de Bordeaux · 2015
In this paper we study the expansions of real numbers in positive and negative real base as introduced by Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets ℤ β + and ℤ - β of nonnegative β -integers and ( - β ) -integers. We describe all bases ( ± β ) for which ℤ β + and ℤ - β can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for β with another interesting property, namely that any linear combination of non-negative powers of the base - β with coefficients in { 0 , 1 , ⋯ , ⌊ β ⌋ } is a ( - β ) -integer, although the corresponding sequence of digits is forbidden as a ( - β ) -expansion.