Rauzy Fractals and their Number-Theoretic Applications

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...

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Bibliographic Details
Published inJournal of mathematical sciences (New York, N.Y.) Vol. 260; no. 2; pp. 265 - 274
Main Author Shutov, A. V.
Format Journal Article
LanguageEnglish
Published New York Springer US 2022
Springer
Springer Nature B.V
Subjects
Fractals
Mathematics
Mathematics and Statistics
Tiles
Toruses
numeral system
11Kxx
Rauzy partition
set of bounded remainder
additive problem
28A80
and phrases
11J71
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Summary: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.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-022-05690-6