O’Bryant, Kevin Fraenkel’s partition and Brown’s decomposition. (English) Zbl 1049.11025 Integers 3, Paper A11, 17 p. (2003). 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 \({\mathbb R}/{\mathbb Z}\). Reviewer: Štefan Porubský (Praha) Cited in 4 Documents MSC: 11B83 Special sequences and polynomials 11B25 Arithmetic progressions 11B34 Representation functions 11A67 Other number representations Keywords:Ostrowski expansion; Zeckendorf expansion; Beatty sequence; continued fraction; Sturmian sequence Citations:Zbl 0172.32501; Zbl 0804.11021 PDFBibTeX XMLCite \textit{K. O'Bryant}, Integers 3, Paper A11, 17 p. (2003; Zbl 1049.11025) Full Text: arXiv EuDML