[論文レビュー] Analysis of the Density of Words under Morphism $\{a,b\}$

Jasem Hamoud, Duaa Abdullah|arXiv (Cornell University)|Jan 5, 2026

semigroups and automata theory被引用数 0

ひとこと要約

この論文はフィボナッチ語の密度とそれのモルフィック導出物をモルフィックおよび機械的（Beatty）記述を用いて分析し、密度が黄金比に結びつくことを示し、フィボナッチ語からの非可換不変量を導入する。

ABSTRACT

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.

研究の動機と目的

• {Motivate and study density properties of the Fibonacci word and words derived from it via morphisms over {a,b}.}
• Provide a unified morphic and mechanical description to obtain explicit symbol densities and sharp discrepancy bounds for prefixes.
• Extend density analysis to Fibonacci-like word families and framed variants.
• Introduce a novel noncommutative invariant in Z<langle a,b rangle> built from finite Fibonacci words and analyze its independence of index.

提案手法

• {Use the classical morphism fixed point 0→01, 1→0 to generate the Fibonacci word and relate it to Beatty sequences for a mechanical representation.}
• {Derive explicit symbol densities from Beatty sequences: delta_0(F)=1/phi and delta_1(F)=1/phi^2.}
• {Obtain O(1) discrepancy bounds for finite prefixes of F.}
• {Generalize density analysis to Fibonacci-like words over {a,b} including framed variants and compute asymptotic densities (also tied to phi).}
• {Define and analyze a power operation in the free noncommutative algebra Z⟨a,b⟩ and prove an index-independent invariant Pow(F_k).}

実験結果

リサーチクエスチョン

• RQ1 {What are the densities of symbols in the infinite Fibonacci word and its morphic/mechanical representations?}
• RQ2 {How do densities behave for derived Fibonacci-like words over {a,b}, including framed constructions?}
• RQ3 {Can a noncommutative invariant be formed from finite Fibonacci words that remains independent of index?}
• RQ4 {What are the finite-prefix discrepancy bounds for symbol counts in the Fibonacci word?}
• RQ5 {How do Beatty sequences and Sturmian dynamics relate to density properties in these words?}

主な発見

• {The densities of 0s and 1s in the infinite Fibonacci word exist and are delta_0(F)=1/phi and delta_1(F)=1/phi^2.}
• {Finite prefixes have counts of 1s approximated by n/phi^2 with O(1) discrepancy and 0s by n/phi with O(1) discrepancy.}
• {Locally, F contains no 11 and every length-3 factor has exactly one 1 and two 0s, aligning with Sturmian and morphic structure.}
• {Framed Fibonacci-like word sequences over {a,b} have asymptotic densities dens_a(Q)=1/phi and dens_b(Q)=1/phi^2, matching the base densities.}
• {A noncommutative invariant Pow(F_k) in Z⟨a,b⟩, defined via a power expression, is independent of k for k≥2 (Pow(F_k) = Pow(F_{k+1}).}
• {The a-density of finite Fibonacci words F_k tends to phi−1 (which equals 1/phi) as k→∞.}