ON PANSIOT WORDS AVOIDING 3-REPETITIONS

On Pansiot Words Avoiding 3-Repetitions

The recently confirmed Dejean’s conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k ≥ 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylin...

Words avoiding repetitions in arithmetic progressions

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi’s construction and the paperfolding words. We extend Carpi’s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avo...

On Maximal Repetitions in Words

Axel Thue's Work on Repetitions in Words

The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

On primary and secondary repetitions in words

Combinatorial properties of maximal repetitions (runs) in formal words are studied. We classify all maximal repetitions in a word as primary and secondary where the set of all primary repetitions determines all the other repetitons in the word. Essential combinatorial properties of primary repetitions are established.