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Threshold languages, which are the (k/(k−1)) + -free languages over k-letter alphabets with k ≥ 5, are the minimal infinite power-free languages according to Dejean's conjecture, which is now proved for all alphabets. We study the growth properties of these languages. On the base of obtained structural properties and computerassisted studies we conjecture that the growth rate of complexity of the threshold language over k letters tends to a constantα ≈ 1.242 as k tends to infinity. Mathematicsdoi:10.1051/ita/2010012 fatcat:6jhxdhlu3rgq5fxxgtkm773gwq