The fractional part of \(n\theta + \phi\) and Beatty sequences. (English) Zbl 0849.11027

Komatsu, Takao

J. Théor. Nombres Bordx. 7, No. 2, 387-406 (1995).

Let \(\theta\), \(\varphi\) be real numbers and \(\{ \cdot \}\), \(|\cdot |\) the fractional part and the distance function from the nearest integer. In the paper explicit formulas for best one-sided approximations to \(x\theta+ \varphi\) are given, i.e. formulas for \(\max_{0\leq x< q_n} \{x\theta+ \varphi\}\), \(\min_{0\leq x< q_n} \{x\theta+ \varphi\}\), \(\max_{0\leq x< q_n} |x\theta+ \varphi |\). The formulas involve the denominator \(q_n\) of the \(n\)th convergent of the continued fraction expansion of \(\theta\) and extend corresponding results proved by T. van Ravenstein [J. Aust. Math. Soc., Ser. A 45, 360-370 (1988; Zbl 0663.10039)] for the homogeneous case \(\varphi =0\). Also modifications are given for the case when \(x=0\) is disallowed. Applications of the proved results to the sorting problem for the fractional part of \(x\theta+ \varphi\) and to finding the characteristic word of the inhomogeneous Beatty sequence \(\lfloor n\theta+ \varphi \rfloor\) are then shown.

Reviewer: Š.Porubský (Praha)

MSC:

11B83	Special sequences and polynomials
11A55	Continued fractions
11B85	Automata sequences

Keywords:

one-sided approximations; continued fraction; sorting problem; characteristic word; Beatty sequence

Citations:

Zbl 0663.10039

References:

[1]	Borwein, J.M. and Borwein, P.B., On the generating function of the integer part: [nα + γ], J. Number Theory43 (1993), 293-318. · Zbl 0778.11039
[2]	Brown, T.C., Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull36 (1993), 15-21. · Zbl 0804.11021
[3]	Danilov, L.V., Some class of transcendental numbers, Mat. Zametki12 (1972), 149-154= 12 (1972), 524-527. · Zbl 0253.10026
[4]	Fraenkel, A.S., Mushkin, M. and Tassa, U., Determination of [nθ] by its sequence of differences, Canad. Math. Bull.21 (1978), 441-446. · Zbl 0401.10018
[5]	Sh., Ito and Yasutomi, S., On continued fractions, substitutions and characteristic sequences [nx + y] - [(n - 1)x + y], Japan. J. Math.16 (1990), 287-306. · Zbl 0721.11009
[6]	Komatsu, T., A certain power series associated with Beatty sequences, manuscript. · Zbl 0858.11013
[7]	Komatsu, T., On the characteristic word of the inhomogeneous Beatty sequence, Bull. Austral. Math. Soc.51 (1995), 337-351. · Zbl 0829.11012
[8]	Nishioka, K., Shiokawa, I. and Tamura, J., Arithmetical properties of certain power series, J. Number Theory42 (1992), 61-87. · Zbl 0770.11039
[9]	Van Ravenstein, T., The three gap theorem (Steinhaus conjecture), J. Austral. Math. Soc. (Series A) 45 (1988), 360-370. · Zbl 0663.10039
[10]	Van Ravenstein, T., Winley, G. and Tognetti, K., Characteristics and the three gap theorem, Fibonacci Quarterly28 (1990), 204-214. · Zbl 0709.11011
[11]	Stolarsky, K.B., Beatty sequences, continued fractions, and certain shift operators, Canad. Math. Bull.19 (1976), 473-482. · Zbl 0359.10028
[12]	Venkov, B.A., Elementary Number Theory, Wolters-Noordhoff, Groningen, 1970. · Zbl 0204.37101