Gowers norms for the Thue–Morse and Rudin–Shapiro sequences
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1897-1913

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.

Received : 2017-03-24
Revised : 2018-02-17
Accepted : 2018-06-12
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3285
Classification:  11B85,  11B30
Keywords: Gowers norm, automatic sequence, Thue–Morse sequence, Rudin–Shapiro sequence
@article{AIF_2019__69_4_1897_0,
     author = {Konieczny, Jakub},
     title = {Gowers norms for the Thue--Morse and Rudin--Shapiro sequences},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1897-1913},
     doi = {10.5802/aif.3285},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1897_0}
}
Gowers norms for the Thue–Morse and Rudin–Shapiro sequences. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1897-1913. doi : 10.5802/aif.3285. https://aif.centre-mersenne.org/item/AIF_2019__69_4_1897_0/

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