Generalized Beatty sequences and complementary triples

Abstract

A generalized Beatty sequence is a sequence $V$ defined by $V\left(n\right)=p\lfloor n\alpha \rfloor +qn+r$, for $n=1,2,\dots$, where $\alpha$ is a real number, and $p,q,r$ are integers. Such sequences occur, for instance, in homomorphic embeddings of Sturmian languages in the integers.

We consider the question of characterizing pairs of integer triples $\left(p,q,r\right),\left(s,t,u\right)$ such that the two sequences $V\left(n\right)=\left(p\lfloor n\alpha \rfloor +qn+r\right)$ and $W\left(n\right)=\left(s\lfloor n\alpha \rfloor +tn+u\right)$ are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that $\alpha$ is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.

We also study triples of sequences $V_i=\left(p_i\lfloor n\alpha \rfloor +q_{i}n+r_i\right)$, $i=1,2,3$ that are complementary in the same sense.