Generalized beatty sequences and complementary triples

Abstract

A generalized Beatty sequence is a sequence V defined by (Formula Presented) where α 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 (p, q, r), (s, t, u) such that the two sequences (Formula Presented) are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that α is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals. We also study triples of sequences (Formula Presented) that are complementary in the same sense.