Digital expansions with negative real bases

TL;DR

This paper characterizes the sequences that appear as negative base expansions of a specific number, extending the understanding of digital expansions to negative real bases and their relation to linear transformations.

Contribution

It provides a new characterization of sequences as negative base expansions of a particular number, generalizing previous work on positive bases and connecting to linear mod one transformations.

Findings

Characterization of sequences as $(-\beta)$-expansions of $\frac{-\beta}{\beta+1}$

Extension of Parry's characterization to negative bases

Connection between expansions and linear transformations with negative slope

Abstract

Similarly to Parry's characterization of $\beta$-expansions of real numbers in real bases $\beta > 1$, 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 $\beta > 1$. In the same vein, we characterize the sequences that occur as $(-\beta)$-expansion of $\frac{-\beta}{\beta+1}$ for some $\beta > 1$. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.