The Undirected Repetition Threshold

Abstract

For rational 1 < r ≤ 2, an undirected r-power is a word of the form xyx ′ , where x is nonempty, x ′ ∈ {x, x R }, and |xyx ′ |/|xy| = r. The undirected repetition threshold for k letters, denoted URT(k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that URT(3) = 7 4 . Then we show that URT(k) ≥ k−1 k−2 for all k ≥ 4. We conjecture that URT(k) = k−1 k−2 for all k ≥ 4, and we confirm this conjecture for k ∈ {4, 8, 12}.

Asymptotic repetitive threshold of balanced sequences

“…For the class of binary sequences rich in palindromes, where RT (C 2 ) = 2 + √ 2 2 ∼ 2.707, this problem is also partially solved: Currie, Mol and Rampersad [10] described all rich sequences with the critical exponent smaller than 2.8. This is in contrast to the results for general sequences.…”

“…Sequences rich in palindromes have not been sufficiently studied and characterized so far. Despite of this fact, Currie, Mol and Rampersad [10] determined the values of the repetition threshold and the asymptotic repetition threshold for the binary alphabet. If C 2 = the set of all binary sequences rich in palindromes, then…”