Gowers norms for automatic sequences

Abstract: We show that any automatic sequence can be separated into a structured part and a Gowers uniform part in a way that is considerably more efficient than guaranteed by the Arithmetic Regularity Lemma. For sequences produced by strongly connected and prolongable automata, the structured part is rationally almost periodic, while for general sequences the description is marginally more complicated. In particular, we show that all automatic sequences orthogonal to periodic sequences are Gowers uniform. As an application, we obtain for any $l \geq 2$ and any automatic set $A \subset \mathbb{N}_0$ lower bounds on the number of $l$-term arithmetic progressions - contained in $A$ - with a given difference. The analogous result is false for general subsets of $\mathbb{N}_0$ and progressions of length $\geq 5$.