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arXiv · 2603.10733

The complexity of smooth words over binary alphabets

Julien Cassaigne·Raphaël Henry
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Abstract

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the emblematic example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $\Theta\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.

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Julien Cassaigne, Raphaël Henry. 2026-03-11. The complexity of smooth words over binary alphabets. https://arxiv.org/abs/2603.10733

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