Further applications of a power series method for pattern avoidance.
Binary words containing infinitely many overlaps.
For each there is an infinite binary word with critical exponent .
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.
Multi-dimensional sets recognizable in all abstract numeration systems
We prove that the subsets of that are -recognizable for all abstract numeration systems are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting.
Multi-dimensional sets recognizable in all abstract numeration systems
We prove that the subsets of that are -recognizable for all abstract numeration systems are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting.
Squares and overlaps in the Thue-Morse sequence and some variants
We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are -regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of ) contains an overlap.
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