On modulo-recurrence and window complexity in infinite words

In this paper, we introduce the notions of uniform modulo-recurrence and strong modulo-recurrence. We show that Sturmian words and maximal complexity words are strongly modulo-recurrent. Next, a relationship between the window complexity and the classical complexity of the Thue-Morse word is established. We provide an aperiodic recurrent word such that the window complexity is bounded. We also construct a family of aperiodic recurrent words such that their window complexity $P^w(n)$ is in $O(n^{\alpha })$, while at least $n^{\alpha }$ for infinitely many $n$, where $0<\alpha <1$. Finally, we establish that the window complexity of a uniformly recurrent aperiodic word is unbounded.