Infinite Words with Well Distributed Occurrences

Abstract

In this paper we introduce the well distributed occurrences (WDO) combinatorial property for infinite words, which guarantees good behavior (no lattice structure) in some related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WDO property if, for each factor w of u, positive integer m, and vector v $\in{\mathbb Z}_{m}^{d}$, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WDO property.