Descriptions of the characteristic sequence of an irrational. (English) Zbl 0804.11021

Brown, Tom C.

Can. Math. Bull. 36, No. 1, 15-21 (1993).

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.

Reviewer: Š.Porubský (Praha)

MSC:

11B83

Special sequences and polynomials

Keywords:

Sturmian sequence; characteristic sequence; Beatty sequence; greatest integer function

Citations:

Zbl 0246.10005