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#### Overlap-Free Words and Generalizations

(University of Winnipeg, 2007)

The study of combinatorics on words dates back at least to the beginning of the
20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains ...

#### Dejean's conjecture holds for n ≥ 27

(EDP Sciences, 2009)

We show that Dejean’s conjecture holds for n ≥ 27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.

#### The Number of Ternary Words Avoiding Abelian Cubes Grows Exponentially

(2004-06-19)

We show that the number of ternary words of length n avoiding abelian cubes grows
faster than r^n, where r = 2^{1/24}

#### Binary Words Containing Infinitely Many Overlaps

(The Electronic Journal of Combinatorics, 2006-09-22)

We characterize the squares occurring in infinite overlap-free binary words and construct various α power-free binary words containing infinitely many overlaps.

#### For each a > 2 there is an Infinite Binary Word with Critical Exponent a

(The Electronic Journal of Combinatorics, 2008-08-31)

The critical exponent of an infinite word w is the supremum of all rational numbers α such that w contains an α-power. We resolve an open question of Krieger and Shallit by showing that for each α>2 there is an infinite ...

#### Squares and overlaps in the Thue-Morse sequence and some variants

(EDP Sciences, 2006)

We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the ...