Gowers norm

"Uniformity norm" redirects here. For the function field norm, see uniform norm. For uniformity in topology, see uniform space.

In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norm on functions on a finite group or group-like object which are used in the study of arithmetic progressions in the group. It is named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.^[1]

Definition

Let f be a complex-valued function on a finite Abelian group G and let J denote complex conjugation. The Gowers d-norm is

\Vert f \Vert_{U^d(G)}^{2^d} = \mathbf{E}_{x,h_1,\ldots,h_d \in G} \prod_{\omega_1,\ldots,\omega_d \in \{0,1\}} J^{\omega_1+\cdots+\omega_d} f\left({x + h_1\omega_1 + \cdots + h_d\omega_d}\right) \ .

Gowers norms are also defined for complex valued functions f on a segment [N]={0,1,2,...,N-1}, where N is a positive integer. In this context, the uniformity norm is given as $\Vert f \Vert_{U^d[N]} = \Vert \tilde{f} \Vert_{U^d(\mathbb{Z}/\tilde{N}\mathbb{Z})}/\Vert 1_{[N]} \Vert_{U^d(\mathbb{Z}/\tilde{N}\mathbb{Z})}$ , where $\tilde N$ is a large integer, $1_{[N]}$ denotes the indicator function of [N], and $\tilde f(x)$ is equal to $f(x)$ for $x \in [N]$ and $0$ for all other $x$ . This definition does not depend on $\tilde N$ , as long as $\tilde N > 2^d N$ .

Inverse conjectures

An inverse conjecture for these norms is a statement asserting that if a bounded function f has a large Gowers d-norm then f correlates with a polynomial phase of degree d-1 or other object with polynomial behaviour (e.g. a (d-1)-step nilsequence). The precise statement depends on the Gowers norm under consideration.

The Inverse Conjecture for vector spaces over a finite field $\mathbb F$ asserts that for any $\delta > 0$ there exists a constant $c > 0$ such that for any finite dimensional vector space V over $\mathbb F$ and any complex valued function $f$ on $V$ , bounded by 1, such that $\Vert f \Vert_{U^{d}[V]} \geq \delta$ , there exists a polynomial sequence $P \colon V \to \mathbb{R}/\mathbb{Z}$ such that

\left| \frac{1}{|V|} \sum_{x \in V} f(x) e(-P(x))\right| \geq c ,

where $e(x) := e^{2 \pi i x}$ . This conjecture was proved to be true by Bergelson, Tao, and Ziegler.^[2]^[3]^[4]

The Inverse Conjecture for Gowers $U^{d}[N]$ norm asserts that for any $\delta > 0$ , a finite collection of (d-1)-step nilmanifolds $\mathcal{M}_\delta$ and constants $c ,C$ can be found, so that the following is true. If $N$ is a positive integer and $f\colon [N]\to \mathbb{C}$ is bounded in absolute value by 1 and $\Vert f \Vert_{U^{d}[N]} \geq \delta$ , then there exists a nilmanifold $G/\Gamma \in \mathcal{M}_\delta$ and a nilsequence $F(g^nx)$ where $g \in G,\ x \in G/\Gamma$ and $F\colon G/\Gamma \to \mathbb{C}$ bounded by 1 in absolute value and with Lipschitz constant bounded by $C$ such that:

\left| \frac{1}{N} \sum_{n =0}^{N-1} f(n) \overline{ F(g^nx}) \right| \geq c .

This conjecture was proved to be true by Green, Tao, and Ziegler.^[5]^[6] It should be stressed that the appearance of nilsequences in the above statement is necessary. The statement is no longer true if we only consider polynomial phases.

References

↑ Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079.
↑ Bergelson, Vitaly; Tao, Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of $\mathbb{F}_p^\infty$ ". Geom. Funct. Anal. 19 (6): 1539–1596. doi:10.1007/s00039-010-0051-1. MR 2594614.
↑ Tao, Terence; Ziegler, Tamar (2010). "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle". Analysis & PDE 3 (1): 1–20. doi:10.2140/apde.2010.3.1. MR 2663409.
↑ Tao, Terence; Ziegler, Tamar (2011). "The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic". Annals of Combinatorics 16: 121–188. doi:10.1007/s00026-011-0124-3. MR 2948765.
↑ Green, Ben; Tao, Terence; Ziegler, Tamar (2011). "An inverse theorem for the Gowers $U^{s+1}[N]$ -norm". Electron. Res. Announc. Math. Sci. 18: 69–90. arXiv:1006.0205. doi:10.3934/era.2011.18.69. MR 2817840.
↑ Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers $U^{s+1}[N]$ -norm". Annals of Mathematics 176 (2): 1231–1372. arXiv:1009.3998. doi:10.4007/annals.2012.176.2.11. MR 2950773.

Tao, Terence (2012). Higher order Fourier analysis. Graduate Studies in Mathematics 142. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8986-2. MR 2931680. Zbl 1277.11010.

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