RAIRO-Theor. Inf. Appl.
Volume 50, Number 1, January-March 2016Special issue dedicated to the 15th "Journées Montoises d'Informatique Théorique"
|Page(s)||93 - 99|
|Published online||02 June 2016|
On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence∗
Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502,
2 CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506 Vandœuvre-lès-Nancy, France
Received: 24 March 2016
Accepted: 24 March 2016
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 ≥ N0(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.
Mathematics Subject Classification: 11A63 / 11B85
Key words: Rudin–Shapiro sequence / automatic sequences / polynomials
© EDP Sciences 2016
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