Dejean's conjecture holds for n>=30

We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n− 3)/6⌋. Let Am = {1, 2, . . . , m}. Let ker ψ = {v ∈ A ∗ m|∀a ∈ Am, 4 divides |v|a}. (In fact, this is not Carpi’s definition of ker ψ, but rather the assertion of his Lemma 9.1.) A word v ∈ A+m is a ψ-kernel repetition if it has period q and a prefix v of length q such that v ∈ ker ψ, (n−1)(|v|+1) ≥ nq − 3. It will be convenient to have the following new definition: If v has period q and its prefix v of length q is in ker ψ, we say that q is a kernel period of v. As Carpi states at the beginning of section 9 of [1]: By the results of the previous sections, at least in the case n ≥ 30, in order to construct an infinite word on n letters avoiding The author is supported by an NSERC Discovery Grant. The author is supported by an NSERC Postdoctoral Fellowship.