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 $\geq 5$.

中文翻译：

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自动序列的 Gowers 范数

我们表明，任何自动序列都可以以比算术正则引理所保证的更有效的方式分成结构化部分和 Gowers 统一部分。对于由强连通和可延展自动机产生的序列，结构部分合理地几乎是周期性的，而对于一般序列，描述稍微复杂一些。特别地，我们表明所有与周期序列正交的自动序列都是高尔一致的。作为一个应用，我们获得任何 $l \geq 2$ 和任何自动集合 $A \subset \mathbb{N}_0$ 的下界 $l$-term 算术级数 - 包含在 $A$ - 中给定的差异。对于 $\mathbb{N}_0$ 的一般子集和长度为 $\geq 5$ 的级数，类似的结果是错误的。