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Article Dans Une Revue Theoretical Computer Science Année : 1999

On repetition-free binary words of minimal density

Roman Kolpakov
  • Fonction : Auteur
Yuri Tarannikov
  • Fonction : Auteur

Résumé

We study the minimal proportion (density) of one letter in $n$-th power-free binary words. First, we introduce and analyse a general notion of minimal letter density for any infinite set of words which don't contain a specified set of ``prohibited'' subwords. We then prove that for $n$-th power-free binary words the density function is $\frac{1}{n}+\frac{1}{n^3}+\frac{1}{n^4}+ {\cal O}(\frac{1}{n^5})$. We also consider a generalization of $n$-th power-free words for fractional powers (exponents): a word is $x$-th power-free for a real $x$, if it does not contain subwords of exponent $x$ or more. We study the minimal proportion of one letter in $x$-th power-free binary words as a function of $x$ and prove, in particular, that this function is discontinuous at $\frac{7}{3}$ as well as at all integer points $n\geq 3$. Finally, we give an estimate of the size of the jumps.

Dates et versions

inria-00098854 , version 1 (26-09-2006)

Identifiants

Citer

Roman Kolpakov, Gregory Kucherov, Yuri Tarannikov. On repetition-free binary words of minimal density. Theoretical Computer Science, 1999, 218 (1), pp.161--175. ⟨10.1016/S0304-3975(98)00257-6⟩. ⟨inria-00098854⟩
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