Geodesic
Greedy and lazy representations in negative base systems
Kybernetika, Tome 49 (2013) no. 2, pp. 258-279 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\beta^2$ with 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 the 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.
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 $\beta^2$ with 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 the 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.
Classification : 11A63, 11A67, 37B10
Keywords: numeration systems; lazy representation; greedy representation; negative base; unique representation 
@article{KYB_2013_49_2_a4,
     author = {Hejda, Tom\'a\v{s} and Mas\'akov\'a, Zuzana and Pelantov\'a, Edita},
     title = {Greedy and lazy representations in negative base systems},
     journal = {Kybernetika},
     pages = {258--279},
     year = {2013},
     volume = {49},
     number = {2},
     mrnumber = {3085396},
     zbl = {1275.11020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2013_49_2_a4/}
}
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Hejda, Tomáš; Masáková, Zuzana; Pelantová, Edita. Greedy and lazy representations in negative base systems. Kybernetika, Tome 49 (2013) no. 2, pp. 258-279. http://geodesic.mathdoc.fr/item/KYB_2013_49_2_a4/