Dejean’s conjecture and Sturmian words

Dejean conjectured that the repetition threshold of a kk-letter alphabet is k/(k−1),k≠3,4k/(k−1),k≠3,4. Values of the repetition threshold for k<5k<5 were found by Thue, Dejean and Pansiot. Moulin-Ollagnier attacked Dejean’s conjecture for 5≤k≤115≤k≤11. Building on the work of Moulin-Ollagnier, we propose a method for deciding whether a given Sturmian word with quadratic slope confirms the conjecture for a given kk. Elaborating this method in terms of directive words, we develop a search algorithm for verifying the conjecture for a given kk. An implementation of our algorithm gives suitable Sturmian words for 7≤k≤147≤k≤14. We prove that for k=5k=5, no suitable Sturmian word exists.