WELLDOC property for words generated by morphisms

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.