# Gowers norm

In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness.[1] They are used in the study of arithmetic progressions in the group. They are named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.[2]

## 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

${\displaystyle \Vert f\Vert _{U^{d}(G)}^{2^{d}}=\mathbf {E} _{x,h_{1},\ldots ,h_{d}\in G}\prod _{\omega _{1},\ldots ,\omega _{d}\in \{0,1\}}J^{\omega _{1}+\cdots +\omega _{d}}f\left({x+h_{1}\omega _{1}+\cdots +h_{d}\omega _{d}}\right)\ .}$

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 ${\displaystyle \Vert f\Vert _{U^{d}[N]}=\Vert {\tilde {f}}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}/\Vert 1_{[N]}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}}$ , where ${\displaystyle {\tilde {N}}}$  is a large integer, ${\displaystyle 1_{[N]}}$  denotes the indicator function of [N], and ${\displaystyle {\tilde {f}}(x)}$  is equal to ${\displaystyle f(x)}$  for ${\displaystyle x\in [N]}$  and ${\displaystyle 0}$  for all other ${\displaystyle x}$ . This definition does not depend on ${\displaystyle {\tilde {N}}}$ , as long as ${\displaystyle {\tilde {N}}>2^{d}N}$ .

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

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

where ${\displaystyle e(x):=e^{2\pi ix}}$ . This conjecture was proved to be true by Bergelson, Tao, and Ziegler.[3][4][5]

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

${\displaystyle \left|{\frac {1}{N}}\sum _{n=0}^{N-1}f(n){\overline {F(g^{n}x}})\right|\geq c.}$

This conjecture was proved to be true by Green, Tao, and Ziegler.[6][7] 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

1. ^ Hartnett, Kevin. "Mathematicians Catch a Pattern by Figuring Out How to Avoid It". Quanta Magazine. Retrieved 2019-11-26.
2. ^ 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. CS1 maint: discouraged parameter (link)
3. ^ Bergelson, Vitaly; Tao, Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of ${\displaystyle \mathbb {F} _{p}^{\infty }}$ ". Geom. Funct. Anal. 19 (6): 1539–1596. arXiv:0901.2602. doi:10.1007/s00039-010-0051-1. MR 2594614.
4. ^ 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. arXiv:0810.5527. doi:10.2140/apde.2010.3.1. MR 2663409.
5. ^ Tao, Terence; Ziegler, Tamar (2011). "The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic". Annals of Combinatorics. 16: 121–188. arXiv:1101.1469. doi:10.1007/s00026-011-0124-3. MR 2948765.
6. ^ Green, Ben; Tao, Terence; Ziegler, Tamar (2011). "An inverse theorem for the Gowers ${\displaystyle U^{s+1}[N]}$ -norm". Electron. Res. Announc. Math. Sci. 18: 69–90. arXiv:1006.0205. doi:10.3934/era.2011.18.69. MR 2817840.
7. ^ Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers ${\displaystyle U^{s+1}[N]}$ -norm". Annals of Mathematics. 176 (2): 1231–1372. arXiv:1009.3998. doi:10.4007/annals.2012.176.2.11. MR 2950773.