Gowers norms for the Thue-Morse and Rudin-Shapiro sequences
Abstract
We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences. The methods can also be extended to other automatic sequences. As an application, we asymptotically count arithmetic progressions in the set of integers ≤ N where the Thue-Morse (resp. Rudin-Shapiro) sequence takes the value +1.
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