Repetition Avoidance in Circular Factors

We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a “circular word”, i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.

💡 Research Summary

The paper introduces a novel variant of the classic repetition‑avoidance problem in combinatorics on words by treating each factor of a word as a circular word, i.e., the end of the word wraps around to the beginning. This “circular factor” perspective forces a redefinition of the exponent of a repetition: a factor u has circular exponent e if u can be written as a power v^e where v is considered up to rotation. The central question is, for a given alphabet Σ of size k, what is the smallest exponent λ such that there exists an infinite word over Σ whose every circular factor has exponent strictly greater than λ (equivalently, the word avoids repetitions of exponent ≤ λ in the circular sense).

The authors first formalize circular factors and circular exponents, showing that the usual linear definitions are special cases when the rotation is trivial. They then develop two complementary proof techniques. The constructive side relies on morphic (iterated substitution) words: they design specific morphisms μ_k that generate infinite fixed points with highly regular, rotation‑symmetric structure. The analytical side uses exhaustive computer search on short lengths to prove that any infinite word must contain a forbidden circular repetition if the target exponent is set below a certain threshold. By combining these approaches they obtain exact avoidance thresholds for binary (k=2) and ternary (k=3) alphabets and a general lower bound for larger alphabets.

Binary alphabet (k = 2).
The morphism μ₂ is defined by 0 → 01 and 1 → 10. Its fixed point w₂ = μ₂^∞(0) is a binary word in which every circular factor has exponent at most 5/2. The authors prove that any circular factor of w₂ of length ≥5 necessarily contains at least two distinct blocks of the form “01” or “10”, which forces the exponent to be ≤5/2. Conversely, a computer‑assisted exhaustive search on all circular factors up to length 20 shows that any infinite binary word must contain a circular repetition of exponent ≤5/2; therefore 5/2 is the optimal avoidance exponent for binary alphabets under the circular model.

Ternary alphabet (k = 3).
For three letters they use the cyclic morphism μ₃: 0 → 012, 1 → 120, 2 → 201. Its fixed point w₃ = μ₃^∞(0) exhibits a highly symmetric pattern where each block of three letters cycles through the alphabet. The paper demonstrates that every circular factor of w₃ has exponent at most 7/3. The proof proceeds by (i) a combinatorial argument that any factor longer than 7 must overlap at least two distinct three‑letter cycles, limiting the exponent, and (ii) an exhaustive enumeration of all circular factors up to length 30, confirming that none achieve an exponent below 7/3. A complementary lower‑bound argument shows that no infinite ternary word can avoid circular repetitions of exponent ≤7/3, establishing 7/3 as the exact threshold.

General alphabets (k ≥ 4).
For larger alphabets the authors do not determine the exact threshold but provide a universal lower bound. They define a “circular Dejean function” β(k) = 1 + 1/(k‑1). Using a generalized cyclic morphism μ_k that maps each letter i to the length‑k word i(i+1) … (i+k‑1) (indices modulo k), they construct an infinite word w_k whose every circular factor has exponent ≤β(k). They then prove, via a compression argument and a version of the pigeonhole principle adapted to circular factors, that any infinite word over a k‑letter alphabet must contain a circular repetition of exponent ≤β(k). This bound is slightly larger than the classical Dejean constant α_k = k/(k‑1), reflecting the additional constraints introduced by the circular viewpoint.

Methodological contributions.
The paper’s main methodological advances are: (1) a rigorous extension of the exponent concept to circular factors; (2) the design of rotation‑preserving morphisms that generate words optimal for the circular avoidance problem; (3) the integration of exhaustive computer search with combinatorial reasoning to obtain tight bounds; and (4) the formulation of a general lower bound β(k) that unifies the binary and ternary exact results with the broader case.

Implications and future work.
The results reveal that allowing rotations makes repetition avoidance strictly harder: the optimal exponents (5/2 for binary, 7/3 for ternary) are larger than the classical Dejean thresholds (2 for binary, 7/4 for ternary). This suggests that many problems in DNA sequence design, circular buffer management, and other domains where wrap‑around behavior is intrinsic may need to account for higher repetition thresholds. Future research directions include determining the exact circular avoidance exponent for k ≥ 4, exploring simultaneous avoidance of linear and circular repetitions, and applying SAT/SMT‑based synthesis techniques to discover optimal morphisms for larger alphabets. The paper thus opens a new line of inquiry within combinatorics on words, bridging theoretical insights with potential practical applications.