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 $${\mathbb R}/{\mathbb Z}$$.

MSC:

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

Citations:

Zbl 0172.32501; Zbl 0804.11021
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