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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 ���� �.
The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
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.
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 powerfre ..."
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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...
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 = 0110100110010110100 ..."
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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
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
NONREPETITIVE COLORINGS OF TREES
"... A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In ..."
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A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that π(T) is bounded by 4 in this class we aim to describe the 4chromatic trees. In particular, we study the 4critical trees which are minimal with respect to this property. Though there are many trees T with π(T) = 4 we show that any of them has a sufficiently large subdivision H such that π(H) = 3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edgecolored by at most ∆ + 1 colors without repetitions on paths.
Infinite words containing squares at every position
 In Proceedings of Journées Montoises D’Informatique Théorique
, 2008
"... Richomme asked the following question: what is the infimum of the real numbers α> 2 such that there exists an infinite word that avoids αpowers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer ..."
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Richomme asked the following question: what is the infimum of the real numbers α> 2 such that there exists an infinite word that avoids αpowers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is α = 7/3. 1
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 p ..."
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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.