On Two Stronger Versions of Dejean’s Conjecture
Repetition threshold is the smallest number RT(n) such that infinitely many n-ary words contain no repetition of order greater than RT(n). These “extremal” repetition-free words are called threshold words. All values of RT(n) are now known, since the celebrated Dejean’s conjecture (1972) was finally settled in 2009. We study two questions about threshold words. First, does the number of n-ary threshold words grow exponentially with length? This is the case for 3 ≤ n ≤ 10, and a folklore conjecture suggests an affirmative answer for all n ≥ 3. Second, are there infinitely many n-ary threshold words containing only finitely many different repetitions of order RT(n)? The answer is “yes” for n = 3, but nothing was previously known about bigger alphabets.
For odd n = 7,9,…,101, we prove the strongest possible result in this direction. Namely, there are exponentially manyn-ary threshold words containing no repetitions of order RT(n) except for the repeats of just one letter.
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