On the Asymptotic Palindrome Density of Fibonacci Infinite Words

TL;DR

This paper explores the palindrome density and combinatorial properties of Fibonacci-generated infinite words, deriving explicit formulas and bounds, and establishing a general density theorem linking sequence structure to the golden ratio.

Contribution

It introduces new results on the density components of Fibonacci-type words and proves a general theorem bounding palindromic prefix density by the inverse of the golden ratio.

Findings

Derived explicit formulas for density components

Established bounds on palindrome densities

Proved a general density theorem involving the golden ratio

Abstract

In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities.