6,212 research outputs found
Lower-bounds on the growth of power-free languages over large alphabets
We study the growth rate of some power-free languages. For any integer
and real , we let be the growth rate of the number
of -free words of a given length over the alphabet .
Shur studied the asymptotic behavior of for as
goes to infinity. He suggested a conjecture regarding the asymptotic
behavior of as goes to infinity when . He
showed that for the asymptotic upper-bound holds of his
conjecture holds.
We show that the asymptotic lower-bound of his conjecture holds. This implies
that the conjecture is true for
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
Syndeticity and independent substitutions
We associate in a canonical way a substitution to any abstract numeration
system built on a regular language. In relationship with the growth order of
the letters, we define the notion of two independent substitutions. Our main
result is the following. If a sequence is generated by two independent
substitutions, at least one being of exponential growth, then the factors of
appearing infinitely often in appear with bounded gaps. As an
application, we derive an analogue of Cobham's theorem for two independent
substitutions (or abstract numeration systems) one with polynomial growth, the
other being exponential
Transition Property for -Power Free Languages with and Letters
In 1985, Restivo and Salemi presented a list of five problems concerning
power free languages. Problem states: Given -power-free words
and , decide whether there is a transition from to . Problem
states: Given -power-free words and , find a transition word
, if it exists.
Let denote an alphabet with letters. Let denote
the -power free language over the alphabet , where
is a rational number or a rational "number with ". If is a "number
with " then suppose and . If is "only" a
number then suppose and or and . We show
that: If is a right extendable word in and
is a left extendable word in then there is a
(transition) word such that . We also show a
construction of the word
- …