论文标题
自动序列的Gowers规范
Gowers norms for automatic sequences
论文作者
论文摘要
我们表明，任何自动序列都可以分为结构化部分，并以比算术规律性引理所保证的效率要高得多的方式。对于通过紧密连接和延长的自动机产生的序列，结构化部分在理性上几乎是周期性的，而对于一般序列，描述的描述更为复杂。特别是，我们表明所有自动序列与周期序列正交均均均匀。作为一个应用程序，我们获得了任何$ l \ geq 2 $和任何自动集合$ a \ subset \ mathbb {n} _0 $下限的$ l $ term arithmetic进度的下限 - 包含在$ a $中，并具有给定差异。对于$ \ mathbb {n} _0 $的一般子集的类似结果和长度$ \ geq 5 $的进度。
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 $\geq 5$.
