Greedy and lazy representations in negative base systems

TL;DR

This paper explores negative base positional systems, providing algorithms for extremal representations, characterizing admissible digit sequences, and analyzing unique representations, especially for special bases like the golden ratio and Tribonacci constant.

Contribution

It introduces algorithms for minimal and maximal representations in negative bases and characterizes admissible digit sequences using positive base representations and forbidden strings.

Findings

Algorithms for extremal representations in negative bases

Characterization of admissible digit sequences

Analysis of unique representations for special bases

Abstract

We consider positional numeration systems with negative real base $-\beta$, where $\beta>1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-\beta)$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base $\beta^2$ and a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy $(-\beta)$-representation. Such a characterization allows us to study the set of uniquely representable numbers. In case that $\beta$ is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy $(-\beta)$-representation using a set of forbidden strings.