  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 $+1$.
