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 be a complex-valued function on a finite abelian group and let denote complex conjugation. The Gowers -norm is

Gowers norms are also defined for complex-valued functions *f* on a segment , where *N* is a positive integer. In this context, the uniformity norm is given as , where is a large integer, denotes the indicator function of [*N*], and is equal to for and for all other . This definition does not depend on , as long as .

## 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 asserts that for any there exists a constant such that for any finite-dimensional vector space *V* over and any complex-valued function on , bounded by 1, such that , there exists a polynomial sequence such that

where . This conjecture was proved to be true by Bergelson, Tao, and Ziegler.^{[3]}^{[4]}^{[5]}

The Inverse Conjecture for Gowers norm asserts that for any , a finite collection of (*d* − 1)-step *nilmanifolds* and constants can be found, so that the following is true. If is a positive integer and is bounded in absolute value by 1 and , then there exists a nilmanifold and a nilsequence where and bounded by 1 in absolute value and with Lipschitz constant bounded by such that:

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

**^**Hartnett, Kevin. "Mathematicians Catch a Pattern by Figuring Out How to Avoid It".*Quanta Magazine*. Retrieved 2019-11-26.**^**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 ".*Geom. Funct. Anal.***19**(6): 1539–1596. arXiv:0901.2602. 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. arXiv:0810.5527. 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. arXiv:1101.1469. doi:10.1007/s00026-011-0124-3. MR 2948765.**^**Green, Ben; Tao, Terence; Ziegler, Tamar (2011). "An inverse theorem for the Gowers -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 -norm".*Annals of Mathematics*.**176**(2): 1231–1372. arXiv:1009.3998. doi:10.4007/annals.2012.176.2.11. MR 2950773.

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