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Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
Fewest repetitions in infinite binary words
A square is the concatenation of a nonempty word with itself. A word has
period p if its letters at distance p match. The exponent of a nonempty word is
the quotient of its length over its smallest period.
In this article we give a proof of the fact that there exists an infinite
binary word which contains finitely many squares and simultaneously avoids
words of exponent larger than 7/3. Our infinite word contains 12 squares, which
is the smallest possible number of squares to get the property, and 2 factors
of exponent 7/3. These are the only factors of exponent larger than 2. The
value 7/3 introduces what we call the finite-repetition threshold of the binary
alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive
threshold
Dejean's conjecture holds for n >= 30
We extend Carpi's results by showing that Dejean's conjecture holds for n >=
30.Comment: introductory material added, minor corrections, 6 page
Finite-Repetition threshold for infinite ternary words
The exponent of a word is the ratio of its length over its smallest period.
The repetitive threshold r(a) of an a-letter alphabet is the smallest rational
number for which there exists an infinite word whose finite factors have
exponent at most r(a). This notion was introduced in 1972 by Dejean who gave
the exact values of r(a) for every alphabet size a as it has been eventually
proved in 2009.
The finite-repetition threshold for an a-letter alphabet refines the above
notion. It is the smallest rational number FRt(a) for which there exists an
infinite word whose finite factors have exponent at most FRt(a) and that
contains a finite number of factors with exponent r(a). It is known from
Shallit (2008) that FRt(2)=7/3.
With each finite-repetition threshold is associated the smallest number of
r(a)-exponent factors that can be found in the corresponding infinite word. It
has been proved by Badkobeh and Crochemore (2010) that this number is 12 for
infinite binary words whose maximal exponent is 7/3.
We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite
word containing only two 7/4-exponent words, the smallest number.
Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question
remains open for alphabets with more than four letters.
Keywords: combinatorics on words, repetition, repeat, word powers, word
exponent, repetition threshold, pattern avoidability, word morphisms.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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