164 research outputs found
Proof of Brlek-Reutenauer conjecture
Brlek and Reutenauer conjectured that any infinite word u with language
closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n)
in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 -
P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity
of u, respectively. This conjecture was verified for periodic words by Brlek
and Reutenauer themselves. Using their results for periodic words, we have
recently proved the conjecture for uniformly recurrent words. In the present
article we prove the conjecture in its general version by a new method without
exploiting the result for periodic words.Comment: 9 page
Continued Fractions of Quadratic Numbers
In this paper, we will first summarize known results concerning continued
fractions. Then we will limit our consideration to continued fractions of
quadratic numbers. The second author described periods and sometimes precise
form of continued fractions of , where is a natural number. In
cases where we were able to find such results in literature, we recall the
original authors, however many results seem to be new.Comment: 13 page
Asymptotic behavior of beta-integers
Beta-integers (``-integers'') are those numbers which are the
counterparts of integers when real numbers are expressed in irrational basis
. In quasicrystalline studies -integers supersede the
``crystallographic'' ordinary integers. When the number is a Parry
number, the corresponding -integers realize only a finite number of
distances between consecutive elements and somewhat appear like ordinary
integers, mainly in an asymptotic sense. In this letter we make precise this
asymptotic behavior by proving four theorems concerning Parry -integers.Comment: 17 page
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
Repetitions in beta-integers
Classical crystals are solid materials containing arbitrarily long periodic
repetitions of a single motif. In this paper, we study the maximal possible
repetition of the same motif occurring in beta-integers -- one dimensional
models of quasicrystals. We are interested in beta-integers realizing only a
finite number of distinct distances between neighboring elements. In such a
case, the problem may be reformulated in terms of combinatorics on words as a
study of the index of infinite words coding beta-integers. We will solve a
particular case for beta being a quadratic non-simple Parry number.Comment: 11 page
Return Words and Recurrence Function of a Class of Infinite Words
Many combinatorial and arithmetical properties have been studied for infinite words ub associated with ß-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.
Factor frequencies in generalized Thue-Morse words
summary:We describe factor frequencies of the generalized Thue-Morse word defined for , as the fixed point starting in of the morphism where and where the letters are expressed modulo . We use the result of Frid [4] and the study of generalized Thue-Morse words by Starosta [6]
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