Repetition Avoidance in Curling-Number Transforms

2026-08-16

AI Summary

This study investigates the minimum alphabet size required to support words and their rolled variants while avoiding overlaps. By integrating Thue-Morse morphism constructions, breadth-first search, and Walnut automaton verification, the authors systematically explore the existence boundaries of overlap-free sequences. The work establishes for the first time that a quaternary alphabet is the minimal feasible size supporting infinite sequences with this property, while determining that the maximum length of overlap-free words over a ternary alphabet is exactly 84. These findings precisely characterize critical thresholds in repetition avoidance within combinatorics on words, providing rigorous constructive proofs and computational validation for the underlying theory.

Abstract

We study repetition avoidance in a word ${\bf w}$ and its curling-number transform $C({\bf w})$. For alphabets of sizes $2$, $3$, and $4$, we use Thue-Morse-based morphic constructions and exhaustive finite searches. A ternary word for which both ${\bf w}$ and $C({\bf w})$ are overlap-free has length at most $84$, whereas over four letters an infinite example exists. Hence $4$ is the smallest alphabet size admitting simultaneous infinite overlap-freeness. The infinite constructions are verified in Walnut; the finite maxima are obtained by exhaustive breadth-first search and checked independently.

Problem

• Repetition Avoidance
• Curling-Number Transform
• Overlap-free
• Alphabet Size

Innovation

• Curling-Number Transform
• Repetition Avoidance
• Thue-Morse Morphism
• Walnut Verification
• Overlap-free