Gowers norms for the Thue–Morse and Rudin–Shapiro sequences

Konieczny, Jakub

Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1897-1913.

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 $+1$.

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 $\le N$ where the Thue–Morse (resp. Rudin–Shapiro) sequence takes the value $+1$.

Classification : 11B85, 11B30
Keywords: Gowers norm, automatic sequence, Thue–Morse sequence, Rudin–Shapiro sequence
Mot clés : Norme de Gowers, suite automatique, suite de Thue–Morse, suite de Rudin–Shapiro

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