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Three distance theorems and combinatorics on words. (English) Zbl 0997.11051

This paper is first a very nice survey on the three distance theorem [see V. Turán-Sos, Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 1, 127-134 (1958; Zbl 0094.02903) and S. Świerczkowski, Fundam. Math. 46, 187-189 (1959; Zbl 0085.27203)] and its generalizations, going from codings of rotations to block complexities of sequences over finite alphabets, and from Beatty sequences to Sturmian sequences. It also contains new results on the frequencies of factors for sequences defined as codings of irrational rotations on \(\mathbb{R}/\mathbb{Z}\). A large bibliography of 58 items is given.
Note that the following papers have appeared:
[4] V. Berthé and L. Vuillon, Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math. 223, 27-53 (2000; Zbl 0970.68124).
[5] J. Anal. Math. 79, 1-31 (1999; Zbl 0996.37006).
[12] Theor. Comput. Sci. 230, 97-116 (2000; Zbl 0947.68543).
[13] J. Théor Nombres Bordx. 13, 371-394 (2001; Zbl 1038.37010).
[22] Theor. Comput. Sci. 215, 31-49 (1999; Zbl 0913.68163).
[23] Acta Arith. 85, 157-177 (1998; Zbl 0910.11007).
[36] SCAN-98 Conference (see Zbl 0949.65013).
[48] Acta Arith. 97, 195-210 (2001; Zbl 1004.11040.

MSC:

11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
68R15 Combinatorics on words
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