Digital expansions with negative real bases

Wolfgang Steiner

doi:10.1007/s10474-012-0252-1

paper

Digital expansions with negative real bases

arXiv:1112.1183 · doi:10.1007/s10474-012-0252-1

Abstract

Similarly to Parry's characterization of $β$ -expansions of real numbers in real bases $β > 1$ , Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base $β > 1$ . In the same vein, we characterize the sequences that occur as $(- β)$ -expansion of $\frac{- β}{β + 1}$ for some $β > 1$ . These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.