Geodesic
Gowers norms for the Thue–Morse and Rudin–Shapiro sequences
[Normes de Gowers pour les suites de Thue–Morse et de Rudin–Shapiro]
Konieczny, Jakub1
1 Mathematical Institute, University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road, Oxford, OX2 6GG (UK)
Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1897-1913.

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We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue–Morse and Rudin–Shapiro sequences. The methods are quite robust and can be extended to a broader class of sequences.

As an application, we asymptotically count arithmetic progressions of a given length in the set of integers ≤N where the Thue–Morse (resp. Rudin–Shapiro) sequence takes the value +1.

Nous estimons les normes de Gowers de suites automatiques classiques telles que les suites de Thue–Morse et de Rudin–Shapiro. Les méthodes utilisées sont assez robustes, et peuvent être étendues à des familles de suites plus générales.

Nous en déduisons une estimation asymptotique du nombre de progressions arithmétiques d’une longueur donnée parmi l’ensemble des indices ⩽N où la suite de Thue–Morse (respectivement, la suite de Rudin–Shapiro) prend la valeur +1.

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DOI : 10.5802/aif.3285
Classification : 11B85, 11B30
Keywords: Gowers norm, automatic sequence, Thue–Morse sequence, Rudin–Shapiro sequence
Mots-clés : Norme de Gowers, suite automatique, suite de Thue–Morse, suite de Rudin–Shapiro

Konieczny, Jakub 1

1 Mathematical Institute, University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road, Oxford, OX2 6GG (UK)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Konieczny, Jakub. Gowers norms for the Thue–Morse and Rudin–Shapiro sequences. Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1897-1913. doi : 10.5802/aif.3285. http://geodesic.mathdoc.fr/articles/10.5802/aif.3285/

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