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Descriptions of the characteristic sequence of an irrational. (English) Zbl 0804.11021
Given a positive irrational real number $$\alpha$$, its characteristic sequence $$f(\alpha)$$ is a $$\{0,1\}$$-sequence defined by $$f(\alpha) = f_ 1f_ 2\dots$$, where $$f_ n = [(n + 1) \alpha] - [n \alpha]$$, $$n \geq 1$$. The author collects and comments on some known procedures which can be used to generate arbitrarily long initial segments of $$f(\alpha)$$. Their description is based on the continued fraction expansion of $$\alpha$$ and uses the language of the theory of homomorphisms of the free group generated by the symbols $$0,1$$. Relations among these descriptions are proved and a simplified proof of an earlier result of A. S. Fraenkel, J. Levitt and M. Shimshoni [Discrete Math. 2, 335- 345 (1972; Zbl 0246.10005)] for an arithmetic expression for $$[m \alpha]$$ involving Zeckendorf representation of $$m-1$$ is given. The paper concludes with some remarks and questions.

##### MSC:
 11B83 Special sequences and polynomials
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