A note on antichains of words.
There are ternary circular square-free words of length for 18.
There exist binary circular power free words of every length.
Binary words containing infinitely many overlaps.
Non-repetitive tilings.
For each there is an infinite binary word with critical exponent .
Extremal infinite overlap-free binary words.
The number of ternary words avoiding abelian cubes grows exponentially.
Infinite words containing squares at every position
Richomme asked the following question: what is the infimum of the real numbers > 2 such that there exists an infinite word that avoids -powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is = 7/3.
Dejean's conjecture holds for N ≥ 27
We show that Dejean's conjecture holds for ≥ 27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.
Least periods of factors of infinite words
We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of sturmian words.
Least Periods of Factors of Infinite Words
We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.
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