Greedy and lazy representations of numbers in the negative golden ratio base

Details Greedy and lazy representations of numbers in the negative golden ratio base Greedy and lazy representations of numbers in the negative golden ratio base

2011-10-28

arxiv.org/abs/1110.6327v1

Mathematics

Dynamical Systems

11 pages, 4 figures

Scientific paper

We consider positional numeration systems with real base (both positive and negative) and study the extremal representations in these systems, called here the greedy and lazy representations. We focus on the base $\beta = - \phi$, where $\phi= \frac{1+\sqrt{5}}{2}$ is the golden mean. We show, that the algorithm introduced by Ito and Sadahiro in 2009 produces neither minimal nor maximal $(-\phi)$-representation with respect to the alternate order and we give algorithms for determination of these extremal strings. We also show that both extremal representations, as well as the Ito-Sadahiro representation, can be obtained using the positive base $\phi^2$ and a non-integer alphabet.

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