Let be an integer-valued polynomial taking only positive values and let be a fixed positive integer. The aim of this short note is to show, by elementary means, that for any sufficiently large integer there exists such that contains exactly occurrences of the block of size in its digital expansion in base . 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.
Accepté le :
DOI : 10.1051/ita/2016009
Keywords: Rudin–Shapiro sequence, automatic sequences, polynomials
Stoll, Thomas 1, 2
@article{ITA_2016__50_1_93_0,
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 = {https://www.numdam.org/articles/10.1051/ita/2016009/}
}
TY - JOUR AU - Stoll, Thomas TI - On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2016 SP - 93 EP - 99 VL - 50 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ita/2016009/ DO - 10.1051/ita/2016009 LA - en ID - ITA_2016__50_1_93_0 ER -
%0 Journal Article %A Stoll, Thomas %T On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2016 %P 93-99 %V 50 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ita/2016009/ %R 10.1051/ita/2016009 %G en %F ITA_2016__50_1_93_0
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
J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003). | Zbl | MR
and , A case study in mathematical research: the Golay–Rudin–Shapiro sequence. Amer. Math. Monthly 103 (1996) 854–869. | Zbl | MR | DOI
and , Congruences de sommes de chiffres de valeurs polynomiales. Bull. London Math. Soc. 38 (2006) 61–69. | Zbl | MR | DOI
and , Newman’s phenomenon for generalized Thue–Morse sequences. Discrete Math. 308 (2008) 1191–1208. | Zbl | MR | DOI
, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13 (1967/1968) 259–265. | Zbl | MR | DOI
M. Lothaire, Applied Combinatorics on Words. Vol. 105 of Encycl. Math. Appl. Cambridge University Press, Cambridge (2005). | Zbl | MR
and , Prime numbers along Rudin–Shapiro sequences. J. Eur. Math. Soc. 27 (2015) 2595–2642. | Zbl | MR | DOI
, On the number of binary digits in a multiple of three. Proc. Amer. Math. Soc. 21 (1969) 719–721. | Zbl | MR | DOI
The Online Encyclopedia of Integer Sequences (OEIS), edited by N.J.A. Sloane. Available at https://oeis.org/. | Zbl
, Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10 (1959) 855–859. | Zbl | MR | DOI
H.S. Shapiro, Extremal Problems for Polynomials and Power Series. Master thesis, M.I.T. (1951). | MR
, The sum of digits of polynomial values in arithmetic progressions. Functiones et Approximatio 47 (2012) 233–239. | Zbl | MR
Cité par Sources :
