On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence

Abstract : Let P (x) ∈ Z[x] be an integer-valued polynomial taking only positive values and let d be a fixed positive integer. The aim of this short note is to show, by elementary means, that for any sufficiently large integer N ≥ N_0(P, d) there exists n such that P(n) contains exactly N occurrences of the block (q − 1, q − 1,... , q − 1) of size d in its digital expansion in base q. The method of proof allows to give a lower estimate on the number of " 0 " resp. " 1 " symbols in polynomial extractions in the Rudin–Shapiro sequence.
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Thomas Stoll. On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), EDP Sciences, 2016, 50 (1), pp.93-99. ⟨10.1051/ita/2016009 ⟩. ⟨hal-01278708⟩

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