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