On possible growth of Toeplitz languages

TL;DR

This paper introduces a new family of factorial languages derived from Toeplitz words, analyzing their subword complexity growth rate using advanced analytical methods, revealing unusual asymptotic behavior.

Contribution

It presents a novel class of languages with complexity growing as a power law, characterized by roots of transcendent equations, and applies Wiener-Pitt theorem-based analysis.

Findings

Subword complexity grows as Θ(n^α) where α is a transcendental root.

Provides asymptotic growth formulas for arithmetical factors of Toeplitz words.

Identifies a new class of languages with unusual complexity growth patterns.

Abstract

We consider a new family of factorial languages whose subword complexity grows as $\Theta(n^{\alpha})$, where $\alpha$ is the root of some transcendent equation. Analytical methods and in particular, a corollary of the Wiener-Pitt theorem, are used to find the asymptotic growth of the complexity. Factorial languages considered are languages of arithmetical factors of some Toeplitz words. So, we describe a new family of words with an unusual growth of arithmetical complexity.