1,049 research outputs found
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
Infinite binary words containing repetitions of odd period
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 its length divided by its smallest period. In this article, we give some new results on the trade-off between the number of squares and the number of cubes in infinite binary words whose square factors have odd periods
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
Complement Avoidance in Binary Words
The complement of a binary word is obtained by changing
each in to and vice versa. We study infinite binary words
that avoid sufficiently large complementary factors; that is, if is a
factor of then is not a factor of . In
particular, we classify such words according to their critical exponents
Antisquares and Critical Exponents
The complement of a binary word is obtained by changing each
in to and vice versa. An antisquare is a nonempty word of the form
. In this paper, we study infinite binary words that do not
contain arbitrarily large antisquares. For example, we show that the repetition
threshold for the language of infinite binary words containing exactly two
distinct antisquares is . We also study repetition thresholds
for related classes, where "two" in the previous sentence is replaced by a
large number.
We say a binary word is good if the only antisquares it contains are and
. We characterize the minimal antisquares, that is, those words that are
antisquares but all proper factors are good. We determine the the growth rate
of the number of good words of length and determine the repetition
threshold between polynomial and exponential growth for the number of good
words
On infinite words avoiding a finite set of squares
Building an infinite square-free word by appending one letter at a time while
simultaneously avoiding the creation of squares is most likely to fail. When
the alphabet has two letters this approach is impossible. When the alphabet has
three or more letters, one will most probably create a word in which the
addition of any letter invariably creates a square. When one restricts the set
of undesired squares to a finite one, this can be possible. We study the
constraints on the alphabet and the set of squares which permit this approach
to work.Comment: 18 page
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