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- Title
On the Three-Distance Theorem.
- Authors
Berthé, Valérie; Reutenauer, Christophe
- Abstract
The given text discusses the Three-Distance Theorem, which states that a sequence of fractional parts of points, together with 1, partitions the unit interval into intervals with at most three different lengths. The text explores the finite words that encode the lengths of these intervals and their relationship to three-interval exchanges. It introduces the concept of symmetric discrete interval exchange and provides examples to illustrate the encoding process. The text also discusses cyclic restrictions in permutations and presents a proof of a theorem. It further discusses the properties of sequences and sub-sequences, as well as the concept of distance encoding and its relationship to sequences of distances. The text concludes by discussing the use of dynamical systems and combinatorial tools in the study of interval exchange transformations.
- Subjects
CONTINUED fractions; IRRATIONAL numbers; EUCLIDEAN algorithm; RATIONAL numbers; PERMUTATIONS; METRIC spaces; ERGODIC theory
- Publication
Mathematical Intelligencer, 2024, Vol 46, Issue 2, p183
- ISSN
0343-6993
- Publication type
Article
- DOI
10.1007/s00283-023-10316-z