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Bounds of asymptotic occurrence rates of some patterns in binary words related to integer-valued logistic maps

Abstract : In this article, we investigate the asymptotic occurrence rates of specific subwords in any infinite binary word. We prove that the asymptotic occurrence rate for the subwords is upper- and lower-bounded in the same way for every infinite binary word, in terms of the asymptotic occurrence rate of the zeros. We also show that both of the bounds are best-possible by constructing, for each bound, a concrete infinite binary word such that the bound is reached. Moreover, we apply the result to analyses of recently-proposed pseudorandom number generators that are based on integer-valued variants of logistic maps.
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Koji Nuida. Bounds of asymptotic occurrence rates of some patterns in binary words related to integer-valued logistic maps. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.709-720. ⟨hal-01185388⟩

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