Generalized Beatty sequences and complementary triples

Abstract

A generalized Beatty sequence is a sequence $V$ defined by $V(n)=p\lfloor{n\alpha}\rfloor+qn +r$, for $n=1,2,\dots$, where $\alpha$ is a real number, and $p,q,r$ are integers. These occur in several problems, as for instance in homomorphic embeddings of Sturmian languages in the integers. Our results are for the case that $\alpha$ is the golden mean, but we show how some results generalise to arbitrary quadratic irrationals. We mainly consider the following question: For which sixtuples of integers $p,q,r,s,t,u$ are the two sequences $V=(p\lfloor{n\alpha}\rfloor+qn +r)$ and $W=(s\lfloor{n\alpha}\rfloor+tn +u)$ complementary sequences? We also study complementary triples, i.e., three sequences $V_i=(p_i\lfloor{n\alpha}\rfloor+q_in+r_i), \:i=1,2,3$, with the property that the sets they determine are disjoint with union the positive integers.