Generalized Beatty sequences and complementary triples

TL;DR

This paper investigates the properties of generalized Beatty sequences, especially for the golden mean, focusing on conditions for pairs and triples of such sequences to be complementary, with some results extending to quadratic irrationals.

Contribution

It characterizes when pairs and triples of generalized Beatty sequences are complementary, particularly for the golden mean, and explores generalizations to quadratic irrationals.

Findings

Characterization of complementary pairs of generalized Beatty sequences.

Conditions for three sequences to form a complementary triple.

Extension of results to quadratic irrational numbers.

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.