Beatty sequences and Langford sequences
A disjoint covering system (DCS) of Beatty sequences is the collection of Beatty sequences \((\alpha_ i,\beta_ i)=\{\lfloor n_ i+ n_ i\rfloor: n\in{\mathbb{Z}}^ +\}\) with \(\alpha_ i>0\), \(\beta_ i\) real numbers, which partition \(\mathbb{Z}^ +\). If all the moduli \(\alpha_ i\) are integers then we obtain a system of arithmetic progressions covering \(\mathbb{Z}^ +\), so called integer DCS's. This fact provides an intuitive link between this notion and the Langford sequences \(C=\{c_ n\}_{n=1}^ \infty\) which are sequences defined by the property that for each of its elements \(c\in C\), the sequence \(\{n: c_ n=c\}\) is an infinite arithmetic progression with common difference \(c\). However, as the authors show, this link leads to a natural connection between Langford sequences and infinite integer DCS's with distinct moduli. To extend this relationship also to certain DCS's with rational moduli, the concept of a quasi-Langford sequence is introduced in the paper and these ties are used as the basis for proof of some properties of quasi- Langford sequences and vice versa. So for instance two conjectures on quasi-Langford sequences are formulated, their equivalence is proved and it is shown that they imply an older conjecture on DCS's with rational moduli.
scientific article; zbMATH DE number 428311
scientific article