On the characteristic word of the inhomogeneous Beatty sequence. (English) Zbl 0829.11012

Komatsu, Takao

Bull. Aust. Math. Soc. 51, No. 2, 337-351 (1995).

The characteristic word \(f_1 f_2 f_3 \ldots\) of the inhomogeneous Beatty sequence \(([n \theta + \varphi])\) of integer parts of \(n \theta + \varphi\), \(n = 1,2, \ldots\) is defined through the 0,1 sequence \(f_1 f_2 f_3 \ldots\) where \(f_n = [(n + 1) \theta + \varphi] - [n \theta + \varphi] - [\theta]\). The author extends earlier constructions for the characteristic word of inhomogeneous Beatty sequences with rational values of \(\varphi\) to arbitrary real \(\varphi\) (and \(\theta)\). The constructions are based on B. A. Venkov’s method [Elementary number theory. Groningen: Wolters-Noordhoff Publishing (1970; Zbl 0204.37101), 65–68] and depend on the continued fraction expansion of \(\theta = [0, a_1, a_2, \ldots]\), and an expansion of \(\varphi\) in terms of the sequence \(\{\theta_0, \theta_1, \ldots\}\), where \(\theta_{n - 1} = [0, a_n, a_{n + 1}, \ldots]\).

Reviewer: Štefan Porubský (Praha)

MSC:

11B83	Special sequences and polynomials
11A55	Continued fractions
11J70	Continued fractions and generalizations

Keywords:

characteristic word; inhomogeneous Beatty sequence

Citations:

Zbl 0204.37101

References:

[1]	Ito, Japan. J. Math. 16 pp 287– (1990)
[2]	Fraenkel, Canad. Math. Bull. 21 pp 441– (1978) · Zbl 0401.10018
[3]	Fraenkel, Canad. J. Math. 21 pp 6– (1969) · Zbl 0172.32501
[4]	Danilov, Mat. Zametki pp 149– (1972)
[5]	Komatsu, J. Théorie des Nombres de Bordeaux
[6]	Venkov, Elementary number theory (1970) · Zbl 0204.37101
[7]	Ravenstein, Fibonacci Quart. 28 pp 204– (1990)
[8]	DOI: 10.1016/0022-314X(92)90109-3 · Zbl 0770.11039
[9]	Stolarsky, Canad. Math. Bull. 19 pp 473– (1976) · Zbl 0359.10028