Letter frequency in infinite repetition-free words

We estimate the extremal letter frequency in infinite words over a finite alphabet avoiding some repetitions. For ternary square-free words, we improve the bounds of Tarannikov on the minimal letter frequency, and prove that the maximal letter frequency is 255/653. Kolpakov et al. have studied the function rho such that rho(x) is the minimal letter frequency in an infinite binary x-free word. In particular, they have shown that rho is discontinuous at 7/3 and at every integer at least 3. We answer one of their questions by providing some other points of discontinuity for rho. Finally, we propose stronger versions of Dejean's conjecture on repetition threshold in which unequal letter frequencies are required. (c) 2007 Elsevier B. V. All rights reserved.