@MISC{Currie09dejean’sconjecture, author = {James Currie and Narad Rampersad}, title = {Dejean’s conjecture holds for n ≥ 27}, year = {2009} }

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Abstract

We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an n-letter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and Mohammad-Noori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.