Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density
Jasem Hamoud, Duaa Abdullah
TL;DR
This work investigates the density and complexity of Fibonacci words and their base-$b$ generalizations within combinatorics on words, bridging symbolic dynamics and number theory. It proves that the infinite word $\mathfrak{F}_{b}$ has full factor and arithmetic complexity, with $p_{\mathfrak{F}_{b}}(k)=b^{k}$ and $a_{\mathfrak{F}_{b}}(k)=b^{k}$, underpinned by equidistribution of base-$b$ factorial representations. A Lucas-number-based density framework is developed via $\mathrm{dens}(p)=\frac{N(p)}{p^{e}}+\frac{Z(p)}{2p^{2e-1}(p+1)}$, linking residue counts modulo $p^{e}$ to combinatorial complexity. These results position Fibonacci words as canonical Sturmian systems with broad implications for symbolic dynamics, automata theory, and algebraic number theory.
Abstract
The paper explores combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. These infinite sequences, measures the diversity of subwords in Fibonacci words, showing non-decreasing growth for infinite sequences. Extends factor analysis to arithmetic progressions of symbols, highlighting generalized pattern distributions. Recent results link Sturmian sequences (including Fibonacci words) to unbounded binomial complexity and gap inequivalence, with implications for formal language theory and automata. This work underscores the interplay between substitution rules, algebraic number theory, and combinatorial complexity in infinite words, providing tools for applications in fractal geometry and theoretical computer science.