Density Characterization with The Upper Bound of Density of Fibonacci Word

TL;DR

The paper investigates natural density and symbol densities in Fibonacci words, defining $\operatorname{DF}(F_k) = n/(n+m)$ with word length $|F_k|=n+m=f_k$. It relates these densities to the golden ratio $\varphi$ and a constant $\kappa$, derives an upper bound $\operatorname{DF}(F_k) < \frac{m(m+1)}{n(2m-n+1)}$, and develops generating-function-based expressions linking combinatorial structure to limit behavior. By analyzing natural density of word constructions and length ratios, it shows that the density of Fibonacci-word patterns can approach unity, connecting to classical Fibonacci-number distributions. The results provide generating-function frameworks and polynomial identities for Fibonacci-word distributions, offering insights into the asymptotic balance between zeros and ones and the impact of concatenation on density. These findings illuminate the deep ties between Sturmian/fibonacci words and density phenomena in number theory and combinatorics on words.

Abstract

This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote the units digit. Through analysis of these ratios and their convergence to powers of $\varphi$, we illustrate the intrinsic exponential growth rates characteristic of Fibonacci words. By considering the natural density concept for sets of positive integers, it is demonstrated that the density of Fibonacci words approaches unity, correlating with classical results on Fibonacci number distributions as \[ \operatorname{DF}(F_k) <\frac{m(m+1)}{n(2m-n+1)}. \] Furthermore, generating functions and combinatorial formulas for general terms of Fibonacci words are derived, linking polynomial expressions and limit behaviors integral to their combinatorial structure. The study is supplemented by numerical data and graphical visualization, confirming theoretical findings and providing insights into the early transient and asymptotic behavior of Fibonacci word densities.

Figures (2)

• Figure 1: Trend of Fibonacci sequence ratios and differences across $n=1$ to $n=24$.
• Figure 2: Application of the natural density.

Theorems & Definitions (22)

• Definition 2.1: Mehraban2006Gulliver
• Definition 2.2: Sturmian sequence Allouche2003Baake
• Lemma 2.1: Mehraban2006Gulliver
• Theorem 2.2: Allouche2003Baake
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.1
• proof