Gowers norms for automatic sequences

TL;DR

This paper demonstrates that automatic sequences can be efficiently decomposed into structured and Gowers uniform parts, with implications for understanding their arithmetic progressions and orthogonality to periodic sequences.

Contribution

It introduces a more efficient decomposition method for automatic sequences into structured and Gowers uniform components, improving upon the Arithmetic Regularity Lemma.

Findings

Automatic sequences can be separated into structured and Gowers uniform parts more efficiently than traditional methods.

Sequences produced by certain automata are rationally almost periodic.

Automatic sequences orthogonal to periodic sequences are Gowers uniform.

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 $\geq 5$.