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Gowers norm

From Wikipedia, the free encyclopedia

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

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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

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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

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  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". Geometric & Functional Analysis. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079. S2CID 124324198.
  3. ^ Bergelson, Vitaly; Tao, Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of ". Geometric & Functional Analysis. 19 (6): 1539–1596. arXiv:0901.2602. doi:10.1007/s00039-010-0051-1. MR 2594614. S2CID 10875469.
  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. S2CID 16850505.
  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. S2CID 253591592.
  6. ^ 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.
  7. ^ 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. S2CID 119588323.