Dejean’s conjecture and Sturmian words

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