Fraenkel’s partition and Brown’s decomposition. (English) Zbl 1049.11025

A. S. Fraenkel’s partition theorem [Can. J. Math. 21, 6–27 (1969; Zbl 0172.32501)] gives a necessary and sufficient condition for two Beatty sequences \(B(\alpha,\alpha^\prime)=\left\{\left\lfloor{n-\alpha^\prime\over\alpha}\right\rfloor\right\}_{n=1}^\infty\), \(i=1,2\), to tile the set of positive integers. T. C. Brown’s decomposition theorem [Cana. Math. Bull. 36, 15–21 (1993; Zbl 0804.11021)] gives a quantitative description of the so called characteristic word of the sequence \(B(\alpha,0)\), \(\alpha\in(0,1)\), in terms of the continued fraction convergents of \(\alpha\). The author gives two new proofs of these theorems. The proofs are based on a characterization of integers \(k\) belonging to \(B(\alpha,\alpha^\prime)\) in terms of the position of the fractional parts of \(k\alpha\) in the natural circular ordering of \({\mathbb R}/{\mathbb Z}\).


11B83 Special sequences and polynomials
11B25 Arithmetic progressions
11B34 Representation functions
11A67 Other number representations
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