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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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Cited by 22 (3 self)
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
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...
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...