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

Stoll, Thomas

doi:10.1051/ita/2016009

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

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 Cet article a éte moissonné depuis la source Numdam

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Résumé

Let $P (x) \in ℤ [x]$ 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 $N \geq N_{0} (P, d)$ 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 : 2016-03-24
Accepté le : 2016-03-24

MR Zbl

DOI : 10.1051/ita/2016009

Classification : 11A63, 11B85
Keywords: Rudin–Shapiro sequence, automatic sequences, polynomials

Affiliations des auteurs :

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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     author = {Stoll, Thomas},
     title = {On digital blocks of polynomial values and extractions in the {Rudin{\textendash}Shapiro} sequence},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {93--99},
     year = {2016},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {1},
     doi = {10.1051/ita/2016009},
     zbl = {1419.11014},
     mrnumber = {3518161},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ita/2016009/}
}

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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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