On the Existence of 1-Bounded Bi-ideals with the WELLDOC Property

A combinatorial condition called well distributedoccurrences, or WELLDOC for short, has been introducedrecently. The proofs that WELLDOC property holds for thefamily of Sturmian words, and more generally, for Arnoux-Rauzy words are given in two papers by Balkova et al. The WELLDOC property for bounded bi-ideals is analysed inthis paper. The existence of a 1-bounded bi-ideal over thefinite alphabet that satisfies the WELLDOC property has beenproved by the authors.