wseas-edit-12189-20261012064202-c2e5d6 20261012064202922 wseas/wseas wseas.group@gmail.com WSEAS International Journal of Computational and Applied Mathematics & Computer Science 2769-2477 06 10 2026 06 10 2026 6 Alessio Drivet GeoGebra Institute of Turin, ITALY 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. 10 12 2026 10 12 2026 86 8 https://creativecommons.org/licenses/by/4.0/ 10.37394/232028.2026.6.8 https://wseas.com/journals/articles.php?id=12189 Berstel, J. (1995). Axel Thue’s papers on repetitions in words: A translation (Publications du LaCIM No. 20). Université du Québec à Montréal. Dejean, F. (1972). Sur un théorème de Thue. Journal of Combinatorial Theory, Series A, 13(1), 90–99. Currie, J. D., & Rampersad, N. (2011). A proof of Dejean’s conjecture. Mathematics of Computation, 80(274), 1063–1070. Rao, M. (2011). Last cases of Dejean’s conjecture. Theoretical Computer Science, 412(27), 3010–3018 Berthé, V., & Rigo, M. (Eds.). (2010). Combinatorics, automata and number theory. Cambridge University Press. Lothaire, M. (2002). Algebraic combinatorics on words. Cambridge University Press. Badkobeh, G., & Crochemore, M. (2011). Finite-repetition threshold for infinite ternary words. Electronic Proceedings in Theoretical Computer Science, 63, 37–43. https://doi.org/10.4204/EPTCS.63.7 Badkobeh, G., & Crochemore, M. (2012). Fewest repetitions in infinite binary words. RAIRO—Theoretical Informatics and Applications, 46(1), 17–31. https://doi.org/10.1051/ita/2011109 Badkobeh, G., Crochemore, M., & Rao, M. (2014). Finite repetition threshold for large alphabets. RAIRO—Theoretical Informatics and Applications, 48(4), 419–430. Currie, J. D., Mol, L., & Rampersad, N. (2020). The repetition threshold for binary rich words. Discrete Mathematics & Theoretical Computer Science, 22(1), Article 6. https://doi.org/10.23638/DMTCS-22-1-6 Camungol, S., & Rampersad, N. (2016). Avoiding approximate repetitions with respect to the longest common subsequence distance. Involve, 9(4), 657–666. Fici, G., & Puzynina, S. (2023). Abelian combinatorics on words: A survey. Computer Science Review, 47, Article 100532. Pansiot, J.-J. (1984). On a conjecture of F. Dejean on repetitions in words. Discrete Applied Mathematics, 7(3), 297–311. Ollagnier, J.-M. (1992). Proof of Dejean’s conjecture for alphabets with 5 to 11 letters. Theoretical Computer Science, 95(2), 187–205. Mohammad-Noori, M., & Currie, J. D. (2007). Dejean’s conjecture and Sturmian words. European Journal of Combinatorics, 28(3), 876–890. Carpi, A. (2007). On Dejean’s conjecture over large alphabets. Theoretical Computer Science, 385(1–3), 137–151. Dvořáková, L., Pelantová, E., Opočenská, D., & Shur, A. M. (2022). On minimal critical exponent of balanced sequences. Theoretical Computer Science, 922, 158–169. Allouche, J.-P., & Shallit, J. (2003). Automatic sequences: Theory, applications, generalizations. Cambridge University Press. Entringer, R. C., Jackson, D. E., & Schatz, J. A. (1974). On nonrepetitive sequences. Journal of Combinatorial Theory, Series A, 16(2), 159–164.