Greedy and lazy representations in negative base systems

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

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