Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro

Abstract

Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro Bulletin de la S. M. F., tome 109 (1981), p. 207-215 <http://www.numdam.org/item?id=BSMF_1981__109__207_0> © Bulletin de la S. M. F., 1981, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

The number of factors in a paperfolding sequence

“…As a corollary we obtain that all generalised Rudin-Shapiro sequences in the sense of [12] (which except for the classical Rudin-Shapiro sequence are different from the sequences studied in [2]) have an ultimately affine complexity.…”

“…We recall that a paperfolding sequence is the sequence of ridges and valleys obtained by unfolding a sheet of paper which has been folded infinitely many times (see [3,8,11,12]). In other words the sequence (w(ra)) n>0 is a paperfolding sequence if and only if u(4n) = 0 (respectively 1),…”

“…(a) The sequences we consider here were introduced in [12] and are denned as follows: if Ui is a paperfolding sequence, one defines Wi by u>,-(0) = 0 n-l W{(n) = 2~] Ui(t) modulo 2, for n ^ 1. t=o available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700011655 [7] Factors in a paperfolding sequence 29…”

On an extremal property of the Rudin‐Shapiro sequence

“…Rudin [11] and H. S. Shapiro [12] both discovered independently the existence of a sequence e = (e(«)) of ±'s such that N-l sup n=0 e(n) exp 2i (1) Their sequence e, called the Rudin-Shapiro sequence has been studied over and over again by various authors in various contexts. We mention only a few among many: Brilhart and Carlitz [2], Brilhart, Erdos and Morton [3], Kahane and Salem [6, p. 134] and Mendes France and Tenenbaum [9]. In this article we shall extend the extremal property (1) and show that the exponential sequence (exp HirnQ) is a special case of a general class M 2 of sequences (f(n)), with \f(n)\ = 1, such that sup I s{n)f{n) ;(2 + v / 2)v / N.…”

Pons, Reed-Muller Codes, and Group Algebras

“…In 1981, Mendes France and Tenenbaum [12] (who had never seen Shapiro's 1951 work [25] but had heard of it only via Rudin's "rediscovery" [23], and also were unaware of the works of Golay and Welti), rediscovered all the Welti rows and named them "paper-folding sequences". This was a work in pure mathematics, related to fractal dimensions of plane curves.…”