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On the Density of Critical Factorizations
 Theor. Inform. Appl
, 2001
"... We investigate the density of critical positions, that is, the ratio between the number of critical positions and the number of all positions of a word, in in nite sequences of words of index one, that is, the period of which is longer than half of the length of the word. ..."
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We investigate the density of critical positions, that is, the ratio between the number of critical positions and the number of all positions of a word, in in nite sequences of words of index one, that is, the period of which is longer than half of the length of the word.
Sums of digits, overlaps, and palindromes
 Discrete Math. & Theoret. Comput. Sci
"... Let ¦¨§�©��� � denote the sum of the digits in the base � representation of �. In a celebrated paper, Thue showed that the infinite word ©� ¦ ¥ ©������������������� � is overlapfree, i.e., contains no subword of the form �������� � , where � is any finite word and � is a single symbol. Let ���� � ..."
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Let ¦¨§�©��� � denote the sum of the digits in the base � representation of �. In a celebrated paper, Thue showed that the infinite word ©� ¦ ¥ ©������������������� � is overlapfree, i.e., contains no subword of the form �������� � , where � is any finite word and � is a single symbol. Let ���� � be integers with ���� � , ���� �. In this paper, generalizing Thue’s result, we prove that the infinite word �¨§� � �� � ��©�¦¨§�©������������� � ���� � is overlapfree if and only if ���� �. We also prove that ��§¨ � � contains arbitrarily long squares (i.e., subwords of the form �� � where � is nonempty), and contains arbitrarily long palindromes if and only if ���� �.
On RepetitionFree Binary Words of Minimal Density
 Theoretical Computer Science
, 1999
"... We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfree binary w ..."
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We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfree binary words the density function is 1 n + 1 n 3 + 1 n 4 + O( 1 n 5 ). We also consider a generalization of nth powerfree words for fractional powers (exponents): a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in xth powerfree binary words as a function of x and prove, in particular, that this function is discontinuous at 7 3 as well as at all integer points n 3. Finally, we give an estimate of the size of the jumps. Keywords: Unavoidable patterns, powerfree words, exponent, minimal density. 1 Introduction One of classical topics of formal language theory and word combinatorics is th...
On Sturmian and episturmian words, and related topics
, 2006
"... Combinatorics on words plays a fundamental role in various fields of mathematics, not to mention its relevance in theoretical computer science and physics. Most renowned among its branches is the theory of infinite binary sequences called Sturmian words, which are fascinating in many respects, havin ..."
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Combinatorics on words plays a fundamental role in various fields of mathematics, not to mention its relevance in theoretical computer science and physics. Most renowned among its branches is the theory of infinite binary sequences called Sturmian words, which are fascinating in many respects, having been studied from combinatorial, algebraic, and geometric points of view. The most wellknown example of a Sturmian word is the ubiquitous Fibonacci word, the importance of which lies in combinatorial pattern matching and the theory of words. Properties of the Fibonacci word and, more generally, Sturmian words have been extensively studied, not only because of their significance in discrete mathematics, but also due to their practical applications in computer imagery (digital straightness), theoretical physics (quasicrystal modelling) and molecular biology. The history of Sturmian words dates back to the astronomer J. Bernoulli III (1772) and, as described in Venkov’s book [38], there also exists some early work by Christoffel (1875) and Markoff (1882). The first detailed investigation of Sturmian words was carried out in 1940 by Morse and Hedlund [33], who studied such words under the framework of symbolic dynamics and, in fact, introduced the term “Sturmian”, named after the mathematician Charles François
Binary words containing infinitely many overlaps
"... We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1 ..."
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We characterize the squares occurring in infinite overlapfree binary words and construct various α powerfree binary words containing infinitely many overlaps. 1
OverlapFree Symmetric D0L words
, 2001
"... Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 011010011001011010010110 ..."
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Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlapfree words was studied e. g. by Fife [8] who described all binary overlapfree infinite words and Seebold [13] who proved that the ThueMorse word is essentially the only binary overlapfree word which is a fixed point of a morphism. Nowadays the theory of overlapfree words is a part of a more general theory of pattern avoidance [5]. J.P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlapfree wo
Trees and Term Rewriting in 1910: On a Paper by Axel Thue
"... Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semiThue systems and his work on the combinatorics of words are wellknown. Here we consider his 1910 paper which contains many notions and ideas about trees, term rewriting and word proble ..."
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Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semiThue systems and his work on the combinatorics of words are wellknown. Here we consider his 1910 paper which contains many notions and ideas about trees, term rewriting and word problems which are surprisingly modern and have later come to play important roles in mathematics, logic, and computer science.
Minimal Letter Frequency in NTh PowerFree Binary Words
 in Mathematical Foundations of Computer Science 1997, Lecture Notes in Comput. Sci., 1295, eds. I. Privara and P. Ru˘zička
, 1997
"... We show that the minimal proportion of one letter in an nth powerfree binary word is asymptotically 1=n. We also consider a generalization of nth powerfree words defined through the notion of exponent: a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. ..."
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We show that the minimal proportion of one letter in an nth powerfree binary word is asymptotically 1=n. We also consider a generalization of nth powerfree words defined through the notion of exponent: a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in an xth powerfree binary word as a function of x and prove, in particular, that this function is discontinuous. 1 Introduction One of classical topics of formal language theory and word combinatorics is the construction of infinite words verifying certain restrictions. A typical restriction is the requirement that the word does not contain a subword of the form specified by some general pattern. Results of this kind find their applications in different areas such as algebra, number theory, game theory (see [12, 16]). The oldest results of this kind, dating back to the beginning of the century, are Thue's famous constructions of infinite squ...
How Many Square Occurrences Must a Binary Sequence Contain?
, 2003
"... Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sucient to construct an in nite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We s ..."
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Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sucient to construct an in nite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that this constant is 0:55080:::.
A Sequence related to that of ThueMorse
"... We study a sequence, c, which encodes the lengths of blocks in the ThueMorse sequence. In particular, we show that the generating function for c is a simple product. Consider the sequence c : c 0 , c 1 , c 2 , c 3 , . . . = 1, 3, 4, 5, 7, 9, 11, 12, 13, . . . defined to be the lexicographically l ..."
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We study a sequence, c, which encodes the lengths of blocks in the ThueMorse sequence. In particular, we show that the generating function for c is a simple product. Consider the sequence c : c 0 , c 1 , c 2 , c 3 , . . . = 1, 3, 4, 5, 7, 9, 11, 12, 13, . . . defined to be the lexicographically least sequence of positive integers satisfying n # c implies 2n / # c. In fact, the lexicographic minimality of c makes it possible to replace the previous "implies" with "if and only if." Equivalently, c is defined inductively by c 0 = 1 and c k+1 = ( c k + 1 if (c k + 1)/2 / # c c k + 2 otherwise (1) for k # 0. This sequence was the focus of a problem of C. Kimberling in the American Mathematical Monthly [?]. (In fact, he looked at the sequence 4c 0 , 4c 1 , 4c 2 , . . .) The solution was given by D. M. Bloom [?]. Our Corollary ?? answers essentially the same question. At the 4e Colloque Series Formelles et Combinatoire Algebrique (Montreal, June 1992) Simon Plou#e and Paul Zi...