Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation

TL;DR

This paper investigates the structure of maximal blocks in morphic and automatic words, establishing algebraic and rational limits for their ratios, and explores implications for the irrationality exponents of related numbers.

Contribution

It proves that the limsup of ratios of maximal block positions is algebraic for morphic words and rational for automatic words, linking combinatorial properties to number theory.

Findings

Limit ratios are algebraic for morphic words.

Limit ratios are rational for automatic words.

Irrationality exponents of certain automatic and morphic numbers are rational or algebraic.

Abstract

Let $\mb w$ be a morphic word over a finite alphabet $\Sigma$, and let $\Delta$ be a nonempty subset of $\Sigma$. We study the behavior of maximal blocks consisting only of letters from $\Delta$ in $\mb w$, and prove the following: let $(i_k,j_k)$ denote the starting and ending positions, respectively, of the $k$'th maximal $\Delta$-block in $\mb w$. Then $\limsup_{k\to\infty} (j_k/i_k)$ is algebraic if $\mb w$ is morphic, and rational if $\mb w$ is automatic. As a result, we show that the same conclusion holds if $(i_k,j_k)$ are the starting and ending positions of the $k$'th maximal zero block, and, more generally, of the $k$'th maximal $x$-block, where $x$ is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).