Gowers norm

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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[edit]

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

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 , 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[edit]

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.[2][3][4]

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.[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[edit]

  1. ^ Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. MR 1844079. doi:10.1007/s00039-001-0332-9. 
  2. ^ 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. MR 2594614. doi:10.1007/s00039-010-0051-1. 
  3. ^ 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. MR 2663409. doi:10.2140/apde.2010.3.1. 
  4. ^ Tao, Terence; Ziegler, Tamar (2011). "The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic". Annals of Combinatorics. 16: 121–188. MR 2948765. doi:10.1007/s00026-011-0124-3. 
  5. ^ Green, Ben; Tao, Terence; Ziegler, Tamar (2011). "An inverse theorem for the Gowers -norm". Electron. Res. Announc. Math. Sci. 18: 69–90. MR 2817840. arXiv:1006.0205Freely accessible. doi:10.3934/era.2011.18.69. 
  6. ^ Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers -norm". Annals of Mathematics. 176 (2): 1231–1372. MR 2950773. arXiv:1009.3998Freely accessible. doi:10.4007/annals.2012.176.2.11.