Geodesic
On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence
Stoll, Thomas1, 2 
1 Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, 54506 Vandœuvre-lès-Nancy, France
2 CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506 Vandœuvre-lès-Nancy, France
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 15th "Journées Montoises d'Informatique Théorique", Tome 50 (2016) no. 1, pp. 93-99

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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.

Reçu le :
Accepté le :
DOI : 10.1051/ita/2016009
Classification : 11A63, 11B85
Keywords: Rudin–Shapiro sequence, automatic sequences, polynomials

Stoll, Thomas 1, 2

1 Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, 54506 Vandœuvre-lès-Nancy, France
2 CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506 Vandœuvre-lès-Nancy, France 
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     title = {On digital blocks of polynomial values and extractions in the {Rudin{\textendash}Shapiro} sequence},
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     publisher = {EDP-Sciences},
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Stoll, Thomas. On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 15th "Journées Montoises d'Informatique Théorique", Tome 50 (2016) no. 1, pp. 93-99. doi: 10.1051/ita/2016009

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