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arXiv · 2606.13744

On modulo-recurrence and window complexity in infinite words

Julien Cassaigne·Idrissa Kaboré
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Abstract

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.

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Julien Cassaigne, Idrissa Kaboré. 2026-06-11. On modulo-recurrence and window complexity in infinite words. https://arxiv.org/abs/2606.13744

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