Repetition Avoidance under Uniform Frequency Constraints: Limits of Sparse Perturbations

We investigate the interaction between repetition avoidance and global distribution constraints in infinite words over a finite alphabet. We ask whether uniform letter frequencies can be imposed without increasing the classical repetition threshold. To address this question, we introduce uniform variants of the repetition threshold and develop a general framework based on sparse perturbations. As a main structural result, we prove that insertions of vanishing density preserve asymptotic letter frequencies, showing that local corrections cannot enforce global uniformity. This identifies an inherent limitation of perturbative approaches. We complement the theoretical analysis with computational experiments on prefixes up to 50 000. These experiments show that high uniformity and controlled repetition can coexist under relaxed thresholds for alphabets of size at least 3, whereas the binary case exhibits a distinct obstruction, with the maximum exponent exceeding the prescribed bound. Taken together, our results suggest that the repetition threshold may be robust under uniformity constraints, while highlighting the fundamentally different roles of local and global methods. We conclude with a set of open problems toward explicit constructions combining optimal repetition bounds with uniform frequencies.