Lower-bounds on the growth of power-free languages over large alphabets
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We study the growth rate of some power-free languages. For any integer k and real β>1, we let α(k,β) be the growth rate of the number of β-free words of a given length over the alphabet {1,2,…, k}. Shur studied the asymptotic behavior of α(k,β) for β≥2 as k goes to infinity. He suggested a conjecture regarding the asymptotic behavior of α(k,β) as k goes to infinity when 1<β<2. He showed that for 9/8≤β<2 the asymptotic upper-bound holds of his conjecture holds. We show that the asymptotic lower-bound of his conjecture holds. This implies that the conjecture is true for 9/8≤β<2.
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