Structure and growth of $\mathbb{R}$-bonacci words

TL;DR

This paper investigates the growth and structure of a class of binary words called $q$-decreasing words, revealing their connection to cutting sequences, and shows that their exponential growth rate varies fractally with $q$, related to Fibonacci-like constants.

Contribution

It establishes a bijective link between $q$-decreasing words and cutting sequence prefixes, and characterizes the growth rate function $\, ext{Phi}(q)$ as strictly increasing, discontinuous at rationals, and fractal.

Findings

Number of $q$-decreasing words grows exponentially with rate $ ext{Phi}(q)$.

$ ext{Phi}(q)$ is strictly increasing and related to Fibonacci constants.

$ ext{Phi}(q)$ exhibits fractal structure and discontinuities at rationals.

Abstract

A binary word is called $q$-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that for any real positive $q$ the number of $q$-decreasing words of length $n$ grows as $C_q \cdot \Phi(q)^n$ for some constant $C_q$ which depends on $q$ but not on $n$. From previous works, it is already known that $\Phi(1)$ is the golden ratio, $\Phi(2)$ is equal to the tribonacci constant, $\Phi(k)$ is $(k+1)$-bonacci constant. We prove that the function $\Phi(q)$ is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function.