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Automata, algebraicity and distribution of sequences of powers. (English, French summaries) Ann. Inst. Fourier (Grenoble) 51 (2001), no. 3, 687–705. This paper studies the distribution of the sequence ({f n})n≥0, where f = ∑ n≥n0 fnxn is a Laurent formal power series with coefficients in the finite field K, and {f} denotes the nonnegative part of f. A generic result is first proved for f ∈ K((x)) having a strictly negative valuation coefficient; the distribution measure is proved to be equivalent to the Haar measure on K [x]. Then it is proved that, when f ∈ K [x], there always exists a continuous distribution if f0 ̸ = 0 and f ̸ = f0. Particular attention is devoted to the case where f is algebraic; the authors recover a result first proved in [J.-P. Allouche and J.-M. Deshouillers, in Colloque de Théorie Analytique des Nombres “Jean Coquet ” (Marseille, 1985), 37–47, Univ. Paris XI, Orsay, 1988; MR0952863 (89i:11088)]; namely, the logarithmic distribution of ({f m})m≥0 exists and its support has Hausdorff dimension zero, when f is algebraic. Furthermore, it is proved in the paper under review that the support of the logarithmic distribution has even sublinear block complexity. The proof is based on the equivalenc

Year: 2010

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