Abstract
In this paper, we construct and study Rauzy partitions of order n for a certain class of Pisot numbers. These partitions are partitions of a torus into fractal sets. Moreover, the action of a certain shift of the torus on partitions introduced is reduced to rearranging the partition tiles. We obtain a number of applications of partitions introduced to the study of the corresponding shift of the torus. In particular, we prove that partition tiles are sets of bounded remainder with respect to the shift considered. In addition, we obtain a number of applications to the study of sets of positive integers that have a given ending of the greedy expansion by a linear recurrent sequence and to generalized Knuth–Matiyasevich multiplications.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 166, Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics,” Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part II, 2019.
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Shutov, A.V. Rauzy Fractals and their Number-Theoretic Applications. J Math Sci 260, 265–274 (2022). https://doi.org/10.1007/s10958-022-05690-6
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DOI: https://doi.org/10.1007/s10958-022-05690-6