Intersecting rational Beatty sequences (Q2855633)

A Beatty sequence has the form \(S(\alpha,\beta)=\{\lfloor n\alpha+\beta\rfloor : n\in {\mathbb Z}\}\). If the modulus \(\alpha=p/q\) is a rational number, the sequence is called rational (for short RBS). It follows from Lemma 1 of \textit{R. J. Simpson} [Discrete Math. 92, No. 1--3, 361--369 (1991; Zbl 0742.11011)] that in an RBS the offset \(\beta\) can be supposed to be an integer. Although an RBS can be regarded as a generalization of an arithmetic progression, the intersection properties of RBS's are more involved. \textit{R. Morikawa} [Bull. Fac. Lib. Arts, Nagasaki Univ., Nat. Sci. 26, No. 1, 1--13 (1985; Zbl 0573.10042)] proved necessary and sufficient conditions for the existence of \(\beta_1\), \(\beta_2\) such that \(S(p_1/q_1,\beta_1)\) and \(S(p_2/q_2,\beta_2)\) are disjoint (for a simplified proof of Morikawa's theorem see [\textit{J. Simpson}, Integers 4, Paper A12, 10 p. (2004; Zbl 1118.11015)]). Call this result the Morikawa condition. In the paper under review the author shows that if the moduli satisfy the Morikawa condition but the sequences do intersect then the consecutive differences take on at most three different values. For the best possible upper bound for the number of consecutive gaps of the intersection of two RBS's not satisfying the Morikawa condition see [\textit{A. S. Fraenkel} and \textit{R. Holzman}, J. Number Theory 50, No. 1, 66--86 (1995; Zbl 0822.11021)].