Cubefree words with many squares

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.

💡 Research Summary

The paper investigates how many distinct squares can appear in infinite binary words that avoid cubes (i.e., are cubefree). After recalling basic definitions—square (xx), cube (xxx), overlap (axaxa)—the authors note that while infinite cube‑free binary words must contain squares, previous constructions (largely based on iterating morphisms, i.e., D0L systems) only yield O(n log n) distinct squares of length n. The goal of the work is to break this barrier and exhibit infinite cubefree binary words that contain exponentially many distinct squares of each length n, and also to prove that for every positive integer n there exists a cubefree binary square of length 2n.

The technical core relies on the Thue–Morse morphism µ (0→01, 1→10). Lemma 3 shows that for every even length m≥6 the Thue–Morse word contains a factor of the form x = 1001 x′ = x′ 1001. Using this structural property, Theorem 6 proves that the word x 0 x 0 is cubefree, while Theorem 7 proves that x 101100 x 101100 is also cubefree. The proofs proceed by exhaustive case analysis on a possible cube yyy inside these words, employing Lemma 4 (a bound on the period of a cube inside an overlap‑free word) and Lemma 5 (a counting argument for occurrences of short factors in a triple repetition). These results together establish Theorem 1: for any n there exists a cubefree binary square of length 2n.

To obtain exponential growth of square counts, Proposition 8 constructs many cubefree binary squares of length n. Starting from any cubefree binary square xx of length 2m, the authors replace some 0’s by a new symbol 2 to obtain a ternary cubefree square yy. A uniform, injective morphism h (0→001011, 1→001101, 2→011001) maps yy back to a binary word while preserving cubefreeness. Because h is uniform, the length multiplies by 12, and at least 2^{m/2} distinct squares are produced, giving an exponential lower bound.

Theorem 2 (the main result) builds an infinite cubefree binary word with exponentially many distinct squares of each length. Let S be a family of cubefree binary squares containing at least 2^{m/2} words of length 12m for every m (as guaranteed by Proposition 8). Take any infinite cubefree word x over the alphabet {2,3}. Interleave the elements of S between the letters of x to form w = x₁S₁x₂S₂⋯. Because each S_i is cubefree and x is cubefree, w remains cubefree. Finally, a uniform injective morphism g (0→001001101, 1→001010011, 2→001101011, 3→011001011) maps w to a binary word g(w) that is still cubefree. The uniformity of g guarantees that the exponential number of distinct squares from the S_i’s survive, so g(w) contains exponentially many distinct squares of length n. Hence an infinite binary cubefree word can have exponential factor complexity.

The paper also notes that the same construction ideas apply to square‑free words over larger alphabets. Proposition 9 shows how to combine an infinite ternary square‑free word with a high‑complexity binary word, then apply a uniform square‑free morphism to obtain an infinite square‑free word with exponential factor complexity.

Overall, the authors demonstrate that the limitation of D0L‑generated cubefree words (O(n log n) squares) is not intrinsic to cubefreeness. By exploiting the regular structure of the Thue–Morse sequence and carefully designed uniform morphisms, they construct infinite binary cubefree words that contain exponentially many distinct squares, and they prove the existence of cubefree binary squares of any even length. This advances the understanding of combinatorial complexity in avoidance languages and opens avenues for further exploration of high‑complexity avoidance sequences.