Avoiding Abelian powers in binary words with bounded Abelian complexity

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden’s theorem, they proved that if a word avoids Abelian $k$-powers for some integer $k$, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian $k$-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian $k$-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb"uhler.

💡 Research Summary

The paper investigates the interplay between Abelian complexity and the occurrence of Abelian powers in infinite binary words. Abelian complexity measures, for each length n, the number of distinct multisets of letters (Abelian types) that appear in factors of that length. Earlier work by the three last authors showed, via van der Waerden’s theorem, that any infinite word avoiding an Abelian k‑power must have unbounded Abelian complexity. This result raises two natural questions: (1) how often do Abelian k‑powers appear in a word whose Abelian complexity is bounded, and (2) does every uniformly recurrent word with bounded Abelian complexity necessarily begin with an Abelian k‑power? The answer is affirmative for several well‑studied families, such as Sturmian words, but the present work demonstrates that the answer is negative in general.

The authors construct a family of uniformly recurrent binary words whose Abelian complexity remains bounded while the words possess infinitely many suffixes that do not start with an Abelian square (the case k = 2). The construction uses a two‑stage morphic substitution scheme. In the first stage a carefully chosen pattern is inserted to restrict the set of possible Abelian types; in the second stage the pattern is iterated, guaranteeing uniform recurrence. By a detailed combinatorial analysis they prove that the set of Abelian types stays within a fixed finite set for all lengths, establishing bounded Abelian complexity, and simultaneously they show that for every integer m there exists a suffix beginning at position m that avoids an Abelian square at its very start. This provides a concrete counter‑example to the conjecture that every uniformly recurrent bounded‑complexity word must begin with an Abelian k‑power.

A second line of results concerns binary overlap‑free words (words avoiding factors of the form axaxa). The authors prove that the shift‑orbit closure of any infinite binary overlap‑free word contains a word whose prefix avoids an Abelian cube (the case k = 3). The proof exploits the well‑known structure of overlap‑free words (e.g., the Thue–Morse word) and shows that by shifting appropriately one can align the word so that the first three blocks never form an Abelian permutation of each other. Hence, even within the highly constrained class of overlap‑free sequences, one can find members that avoid higher‑order Abelian powers at the very beginning.

The paper also studies the effect of morphisms on Abelian complexity. The authors prove a general preservation theorem: if a word has bounded Abelian complexity, then the image of that word under any (not necessarily uniform) morphism also has bounded Abelian complexity. The proof proceeds by bounding the number of possible Abelian types in the image in terms of the maximal length of morphism images and the original bound. This result is significant because it guarantees that many standard constructions—automatic sequences, substitutive sequences, and morphic images—inherit the bounded‑complexity property, allowing the authors to generate further examples and to transfer results between different symbolic dynamical systems.

Finally, the authors connect the long‑standing open problem of whether there exists an infinite binary word avoiding all Abelian squares to two well‑known open questions posed by Pirillo–Varricchio and by Halbeisen–Hungerbühler. They show that the existence of a binary word with bounded Abelian complexity that avoids Abelian squares is equivalent to the existence of a binary word avoiding Abelian squares altogether, and that both formulations are equivalent to the Pirillo–Varricchio conjecture concerning the avoidability of certain patterns in binary sequences, as well as to the Halbeisen–Hungerbühler problem on the minimal size of unavoidable Abelian patterns. This equivalence reframes the Abelian‑square avoidance problem in terms of previously studied combinatorial pattern‑avoidance questions, suggesting that progress on either of the classical problems would immediately resolve the Abelian‑square question.

In summary, the paper makes four major contributions: (i) it provides explicit uniformly recurrent binary words with bounded Abelian complexity that have infinitely many suffixes not starting with an Abelian square; (ii) it shows that the shift‑orbit closure of any binary overlap‑free word contains a word avoiding an Abelian cube at its start; (iii) it establishes that morphic images of bounded‑complexity words preserve bounded Abelian complexity; and (iv) it links the open problem of Abelian‑square avoidance in binary words to two historic conjectures, thereby unifying several strands of research in combinatorics on words. The results deepen our understanding of how quantitative measures of combinatorial richness (Abelian complexity) interact with qualitative avoidance properties (Abelian powers), and they open new avenues for constructing and analyzing low‑complexity sequences with prescribed avoidance characteristics.