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

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.

MSC:

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

Zbl 0663.10039
Full Text:

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.