Repetition Avoidance in Curling-Number Transforms

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.