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Gowers norm

Let ${\displaystyle f}$ be a complex-valued function on a finite abelian group ${\displaystyle G}$ and let ${\displaystyle J}$ denote complex conjugation. The Gowers ${\displaystyle d}$-norm is

${\displaystyle \Vert f\Vert _{U^{d}($G)}^{2^{d}}=\sum _{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 ${\displaystyle [$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 ${\displaystyle {\tilde {N}}>2^{d}N}$.

Aninverse 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.

^ Bergelson, Vitaly; Tao, Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of ${\displaystyle \mathbb {F} _{p}^{\infty }}$". Geometric & Functional Analysis. 19 (6): 1539–1596. arXiv:0901.2602. doi:10.1007/s00039-010-0051-1. MR 2594614. S2CID 10875469.

^ 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.

^ 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. S2CID 119588323.

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