Letter frequency vs factor frequency in pure morphic words

TL;DR

This paper proves that in pure morphic words, the existence of letter frequencies guarantees the existence of all factor frequencies, answering a question posed by Saari.

Contribution

It establishes a new link between letter and factor frequencies in pure morphic words, resolving a previously open question.

Findings

Letter frequencies imply factor frequencies in pure morphic words

Answers a question from Saari's doctoral thesis

Provides a theoretical foundation for frequency analysis in morphic words

Abstract

We prove that, for any pure morphic word $w$, if the frequencies of all letters in $w$ exist, then the frequencies of all factors in $w$ exist as well. This result answers a question of Saari in his doctoral thesis.