This paper studies repetition avoidance simultaneously in an infinite or finite word and in its curling-number transform. For alphabets of sizes 2, 3, and 4, the work combines morphic constructions with exhaustive finite searches and automatic verification.
Among the main results, a ternary word for which both the word and its curling-number transform are overlap-free has length at most 84, while over a four-letter alphabet an infinite example exists. This shows that four is the smallest alphabet size allowing simultaneous infinite overlap-freeness.
Main results
The paper resolves several finite and infinite repetition-avoidance cases across alphabet sizes 2, 3, and 4. Infinite constructions are verified using Walnut, while the finite extremal cases are established through exhaustive searches and independent checks.
My contribution
My work on the project included independently rechecking the main finite-search claims, identifying and correcting several mathematical and definitional issues, and reorganizing substantial parts of the manuscript. In particular, I independently verified the ternary length-84 obstruction, corrected the indexing convention for the curling-number transform and the definition of alpha-plus-free words, and checked the four-letter construction and its transform.
Computational verification
For the ternary extremal cases, I independently reran three distinct exhaustive searches. The overlap-free / overlap-free case gives 6048 words at length 84 and none at length 85; the 9/4-free / overlap-free case gives 10368 at length 84 and none at 85; and the overlap-free / 7/3-free case again gives 6048 at length 84 and none at 85. I also prepared and cleaned supplementary verification files for the project.