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2012 | OriginalPaper | Chapter
On Two Stronger Versions of Dejean’s Conjecture
Authors : Igor N. Tunev, Arseny M. Shur
Published in: Mathematical Foundations of Computer Science 2012
Publisher: Springer Berlin Heidelberg
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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 many
n
-ary threshold words containing
no repetitions of order
RT
(
n
)
except for the repeats of just one letter
.