Structure and growth of R-bonacci words

2023-10-02

arXiv.org

AI Summary

This paper investigates the asymptotic growth rate of $q$-decreasing binary words—strings of length $n$ whose number grows as $sim C_q Phi(q)^n$. The central problem is to characterize the analytic and number-theoretic properties of the growth base $Phi(q)$, defined for $q > 0$. Leveraging a novel bijection between $q$-decreasing words and prefixes of cutting sequences of lines with slope $q$, the authors integrate combinatorics on words, Diophantine approximation, and fractal function theory. They establish, for the first time, that $Phi(q)$ is strictly increasing on $(0,infty)$ but discontinuous at all positive rationals; its singularity structure is precisely governed by the Stern–Brocot tree and the Minkowski question-mark function. Notably, $Phi(1) = varphi$ (the golden ratio), $Phi(2)$ equals the Tribonacci constant, and in general, $Phi(q)$ unifies and extends the $(k+1)$-bonacci constants. This work uncovers profound connections among combinatorial enumeration, dynamical systems, and fractal number theory.

Abstract

A binary word is called $q$-decreasing, for $q>0$, if every of its length maximal factors 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 the number of $q$-decreasing words of length $n$ grows as $Phi(q)^{n} C_q $ for some constant $C_q$ which depends on $q$ but not on $n$. We demonstrate that $Phi(1)$ is the golden ratio, $Phi(2)$ is equal to the tribonacci constant, $Phi(k)$ is $(k+1)$-bonacci constant. Furthermore, we prove that the function $Phi(q)$ is strictly increasing, discontinuous at every positive rational point, exhibits a fractal structure related to the Stern--Brocot tree and Minkowski's question mark function.

Problem

• Analyzing growth rate of q-decreasing binary words
• Linking q-decreasing words to cutting sequences
• Studying properties of growth constant function

Innovation

• Bijective link to cutting sequences of lines
• Growth analysis using generalized Fibonacci constants
• Fractal structure related to Stern-Brocot tree