WELLDOC property for words generated by morphisms

TL;DR

This paper investigates the WELLDOC property, which concerns the distribution of factors in infinite words, and provides a criterion for when words generated by morphisms possess this property.

Contribution

It introduces a criterion to determine the WELLDOC property in infinite words produced by morphisms, expanding understanding of their combinatorial structure.

Findings

Characterization of the WELLDOC property for morphic words

Criterion for WELLDOC based on morphism properties

Application to specific classes of infinite words

Abstract

In this paper, we study an abelian-type property of infinite words called well distributed occurrences, or WELLDOC for short. An infinite word $w$ on a $d$-ary alphabet has the WELLDOC property if, for each factor $u$ of $w$, positive integer $m$, and vector $v\in \mathbb{N}^d$, there is an occurrence of $u$ such that the Parikh vector of the prefix of $w$ preceding such occurrence is congruent to $v$ modulo $m$. The Parikh vector of a finite word $v$ on an alphabet has its $i$-th component equal to the number of occurrences of the $i$-th letter in $v$. We provide a criterion of the WELLDOC property for words generated by morphisms.