Morphic words, Beatty sequences and integer images of the Fibonacci language

TL;DR

This paper explores how morphic words can be decorated to analyze sequences like Beatty sequences and Fibonacci language embeddings, revealing new structures and generalized sequences with integer-valued formulas.

Contribution

It introduces the concept of decorating morphic words to study complex sequences and provides new results on Beatty sequences and Fibonacci language embeddings.

Findings

Decorated morphic words remain morphic, aiding sequence analysis.

Derived generalized Beatty sequences with explicit integer formulas.

Applied to $AA$ sequences and Fibonacci language embeddings.

Abstract

Morphic words are letter-to-letter images of fixed points $x$ of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of $x$. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called $AA$ sequences, where $\alpha$ is a quadratic irrational, $A$ is the Beatty sequence defined by $A(n)=\lfloor \alpha n\rfloor$, and $AA$ is the sequence $(A(A(n)))$. The second example considers homomorphic embeddings of the Fibonacci language into the integers, which turns out to lead to generalized Beatty sequences with terms of the form $V(n)=p\lfloor \alpha n\rfloor+qn+r$, where $p,q$ and $r$ are integers.