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.


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


Zbl 0663.10039
Full Text: DOI Numdam EuDML EMIS


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