Growth of Power-Free Languages over Large Alphabets

We study growth properties of power-free languages over finite alphabets. We consider the function α(k,β) whose values are the exponential growth rates of β-power-free languages over k-letter alphabets and clarify its asymptotic behaviour. Namely, we prove asymptotic formulas for this function for the case β≥2 and suggest such formulas for the case β<2 on the base of some partial results. All obtained formulas correlate very well with the known numerical bounds on the values of α(k,β).