Dejeans conjecture holds for n >= 30

arXiv:0806.0043v2 [math.CO] 6 Jan 2009 Dejean’s conjecture holds for n ≥30 James Currie∗and Narad Rampersad† Department of Mathematics and Statistics University of Winnipeg 515 Portage Avenue Winnipeg, Manitoba R3B 2E9 (Canada) j.currie@uwinnipeg.ca n.rampersad@uwinnipeg.ca November 13, 2018 Abstract We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥30. 1 Introduction Repetitions in words have been studied starting with Thue [12, 13] at the beginning of the previous century. Much study has also been given to repe- titions with fractional exponent [1, 3, 4, 5, 6, 8]. If n > 1 is an integer, then an n-power is a non-empty word xn, i.e., word x repeated n times in a row. For rational r > 1, a fractional r-power is a non-empty word w = x⌊r⌋x′ such that x′ is the prefix of x of length (r −⌊r⌋)|x|. For example, 01010 is a 5/2-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 r-powers? ∗The author is supported by an NSERC Discovery Grant. †The author is supported by an NSERC Postdoctoral Fellowship. 1 We call this infimum the repetitive threshold of an n-letter alphabet, denoted by RT(n). Dejean’s conjecture [4] is that RT(n) =    7/4, n = 3 7/5, n = 4 n/(n −1) n ̸= 3, 4 The values RT(2), RT(3), RT(4) were established by Thue, Dejean and Pansiot, respectively [12, 4, 11]. Moulin-Ollagnier [10] verified Dejean’s con- jecture for 5 ≤n ≤11, while Mohammad-Noori and Currie [9] proved the conjecture for 12 ≤n ≤14. An exciting new development has recently occurred with the work of Carpi [3], who showed that Dejean’s conjecture holds for n ≥33. Verification of the conjecture is now only lacking for a finite number of values. In the present paper, we sharpen Carpi’s methods to show that Dejean’s conjecture holds for n ≥30. 2 Preliminaries The following definitions are from sections 8 and 9 of [3]: 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 [3]: 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 factors of any exponent larger than n/(n−1), it is sufficient to find an infinite word on the alphabet Am avoiding ψ-kernel repetitions. For m = 5, Carpi produces such an infinite word, based on a paper-folding construction. He thus establishes Dejean’s conjecture for n ≥33. In the present paper, we give an infinite word on the alphabet A4 avoiding ψ-kernel repetitions. We thus establish Dejean’s conjecture for n ≥30. 2 Definition 1. Let f : A∗ 4 →A∗ 4 be defined by f(1) = 121, f(2) = 123, f(3) = 141, f(4) = 142. Let w be the fixed point of f. It is useful to note that the frequency matrix of f, i.e., [|f(i)|j]4×4 =   2 1 0 0 1 1 1 0 2 0 0 1 1 1 0 1   has an inverse modulo 4. Remark 1. Let q be a non-negative integer, q ≤1966. Fix n = 32. R1: Word w contains no ψ-kernel repetition v with kernel period q. R2: Word w contains no factor v with kernel period q such that |v|/q ≥ 35/34. Note that 32 31 − 34 31q = 35 34 when q = 342 3 = 385 1 3, so neither piece of the remark implies the other. Note also that the conditions of the remark become less stringent for n = 30, 31. One also verifies that 35 34 + 9 2(1967) ≤32 31 −34 31q for q ≥1967. To show that w contains no ψ-kernel repetitions for n = 30, 31, 32, it thus suffices to verify R1 and to show that word w contains no factor v with kernel period q ≥1967 such that |v|/q ≥35/34 + 9/2(1967). (1) The remarks R1 and R2 are verified by computer search, so we will con- sider the second part of this attack. Fix q ≥1967, and suppose that v is a factor of w with kernel period q, and |v|/q ≥35/34. Without loss of gener- ality, suppose that no extension of v has period q. Write v = sf(u)p where s (resp. p) is a suffix (resp. prefix) of the image of a letter, and |s| ( resp. |p|) ≤2. If |v| ≤q + 2, then 35/34 ≤(q + 2)/q and 1/34 ≤2/q, forcing q ≤68. This contradicts R2. We will therefore assume that |v| ≥q + 3. Suppose |s| = 2. Since |v| ≥q + 3, write v = s0zs0v′, where |s0z| = q. Examining f, we see that the letter as preceding any occurrence of s0 in w is 3 uniquely determined if |s| = 2. It follows that asv is a factor of w with kernel period q, contradicting the maximality of v. We conclude that |s| ≤1. Again considering f, we see that if t is any factor of w of length 3, and u1t, u2t are prefixes of w, then |u1| ≡|u2| (mod 3). Since |v| ≥q +3, we conclude that 3 divides q. W

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