Gap problems for integer part and fractional part sequences
Given a sequence of the form \(\{b_ n\}= \{n\alpha+ \gamma\): \(n\in \mathbb{Z}_ 0\}\) with modulus \(\alpha\geq 1\) and residue \(\gamma\) being real numbers, let \(S(\alpha, \gamma)= \{s_ n\}\) or \(\Omega (\alpha, \gamma)= \{\omega_ n\}\) be the sequences of integers, or fractional parts, resp. of terms of \(\{b_ n\}\). The sequences of type \(S(\alpha, \gamma)\) are called Beatty sequences. Motivated by Steinhaus and Slater gap problems the authors investigate the following related generalized intersection problem: Let \(S= S(\alpha, \gamma)\) and \(T= T(\beta, \delta)\) be two Beatty sequences. Let \(m_ 1< m_ 2<\dots\) be the sequence of terms which appears in both \(S\) and \(T\). Let \(n_ 1< n_ 1< \dots\) and \(k_ 1< k_ 2< \dots\) be the corresponding indices of the subsequences of \(S\) and \(T\), resp. The differences \(n_{i+1}- n_ i\) are called \(S\)-gaps, the differences \(k_{i+1}- k_ i\) are called \(T\)-gaps, and the differences \(m_{i+1}- m_ i\) are called gaps. How many distinct \(S\)- gaps, \(T\)-gaps (so called index gaps) and gaps can be there? A number of interesting theorems on both types of gaps and on their cardinalities is proved. Thus for instance, if one of the moduli is integral, then there are at most three gaps and if there are three, one of them is the sum of the other two. When the moduli of the two Beatty sequences are rational numbers, then the number of gaps, though finite, may be arbitrarily large. The authors find the answer to the question about the best upper bound on the number of gaps as a function of the denominator(s) of the rational modulus (moduli). Further it is shown that the analysis of gaps in the generalized intersection problems can be reduced to a two-dimensional version of the Slater problem. This is used (among other things) to show that in general the intersection of two arbitrary Beatty sequences has finitely many gaps and that the cardinality of both types of gaps can be bounded by a number depending on the moduli \(\alpha\), \(\beta\) of both intersecting sequences.
scientific article; zbMATH DE number 714110