FINITE REPETITION THRESHOLD FOR LARGE ALPHABETS

We investigate the finite repetition threshold for k-letter alphabets, k >= 4, that is the smallest number r for which there exists an infinite r(+)-free word containing a finite number of r-powers. We show that there exists an infinite Dejean word on a 4-letter alphabet (i.e. a word without factors of exponent more than 7/5) containing only two 7/5-powers. For a 5-letter alphabet, we show that there exists an infinite Dejean word containing only 60 5/4-powers, and we conjecture that this number can be lowered to 45. Finally we show that the finite repetition threshold for k letters is equal to the repetition threshold for k letters, for every k >= 6.