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A simple characterization of Sturm numbers. (Une caractérisation simple des nombres de Sturm.) (French) Zbl 0930.11051

A real number is called a Sturm number if it is the slope of a Sturmian sequence (binary coding of a billiard sequence on the square), which is a fixed point of a morphism. These numbers were called “nombres de Sturm” by B. Parvaix [J. Théor. Nombres Bordx. 9, 351-369 (1997; Zbl 0904.11008)], and first studied by D. Crisp, W. Moran, A. Pollington and P. Shiue [J. Théor. Nombres Bordx. 5, 123-137 (1993; Zbl 0786.11041)], who gave the following characterization for the characteristic Sturmian sequences (billiard sequences starting from the origin) that are fixed point of a morphism: they are exactly the sequences with slope \(x:[\alpha_0,\alpha_1,\alpha_2\dots]\) where \[ \begin{alignedat}{3} \text{if }&x>1 &\quad &x=[a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }1\leq a_0\leq a_n\\ \text{if }&0<x<1 &\quad &x=[0,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }a_0\leq a_n\\ \text{or} \text{if }&\tfrac 12<x<1 &\quad &x=[0,1,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }a_0\leq a_n\\ \text{if }&0<x<\tfrac 12 &\quad &x=[0,1,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }1\leq a_0\leq a_n.\end{alignedat} \] In the paper under review a simple arithmetic condition is given: either \(x\) is quadratic with a negative conjugate, or \(x\) is quadratic in \(]0,1[\) with a conjugate outside \(]0,1[\).
Note that “Shiue” is misspelled in Ref. [3].

MSC:

11J70 Continued fractions and generalizations
37B10 Symbolic dynamics
11B85 Automata sequences
68R15 Combinatorics on words
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References:

[1] Berstel, J., Recent results in sturmian words. Developments in Language Theory II, pp. 13-24, World Sci. Publishing, River Edge, NJ, 1996. · Zbl 1096.68689
[2] Berthé, V., Fréquences des facteurs des suites sturmiennes. Theoret. Comput. Sci.165 (1996), 295-309. · Zbl 0872.11018
[3] Crisp, D., Moran, W., Pollington, A., Shuie, P., Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux5 (1993), 123-137. · Zbl 0786.11041
[4] Parvaix, B., Propriétés d’invariance des mots sturmiens. J. Théor. Nombres Bordeaux9 (1997), 351-369. · Zbl 0904.11008
[5] Redmond, D., Number Theory: An Introduction, chapter 4. pp. 210-235. Monographs and Textbooks in Pure and Applied Mathematics, 201. Marcel Dekker, Inc., New York, 1996. · Zbl 0847.11001
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