“Gowers norm”的意思、由来-开放百科全书

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

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

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

, 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

1. ^{{cite journal|authorlink=Timothy Gowers|first=Timothy|last=Gowers|title=A new proof of Szemerédi's theorem|journal=Geom. Funct. Anal.|volume=11|issue=3|pages=465–588|url=http://www.dpmms.cam.ac.uk/~wtg10/sz898.dvi|year=2001|mr=1844079|doi=10.1007/s00039-001-0332-9}}
2. ^{{cite journal | last1=Bergelson | first1=Vitaly | last2=Tao | first2=Terence | author2-link = Terence Tao | last3=Ziegler | first3=Tamar | author3-link = Tamar Ziegler | title=An inverse theorem for the uniformity seminorms associated with the action of

| mr=2594614 | journal=Geom. Funct. Anal. | year=2010 | volume=19 | issue=6 | pages=1539–1596 | doi=10.1007/s00039-010-0051-1}}
3. ^{{cite journal | last1=Tao | first1=Terence | author1-link = Terence Tao | last2=Ziegler | first2=Tamar | author2-link = Tamar Ziegler | title=The inverse conjecture for the Gowers norm over finite fields via the correspondence principle | year=2010 | journal=Analysis & PDE | volume=3 | issue=1 | pages=1–20 | doi=10.2140/apde.2010.3.1 | mr=2663409| arxiv=0810.5527 }}
4. ^{{cite journal| doi = 10.1007/s00026-011-0124-3| title = The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic| journal = Annals of Combinatorics| volume = 16| pages = 121–188| year = 2011| last1 = Tao | first1 = Terence | author1-link = Terence Tao| last2 = Ziegler | first2 = Tamar | author2-link = Tamar Ziegler| mr = 2948765| arxiv = 1101.1469}}
5. ^{{cite journal | last1 = Green | first1 = Ben | author1-link = Ben Green (mathematician)| last2 = Tao | first2 = Terence | author2-link = Terence Tao| last3 = Ziegler | first3 = Tamar | author3-link = Tamar Ziegler| title = An inverse theorem for the Gowers

-norm| journal = Electron. Res. Announc. Math. Sci.| volume = 18| year = 2011| pages = 69–90| doi = 10.3934/era.2011.18.69| mr = 2817840| arxiv = 1006.0205}}
6. ^{{cite journal| doi = 10.4007/annals.2012.176.2.11| title = An inverse theorem for the Gowers

-norm| journal = Annals of Mathematics| volume = 176| issue = 2| pages = 1231–1372| year = 2012| last1 = Green | first1 = Ben | author1-link = Ben Green (mathematician)| last2 = Tao | first2 = Terence | author2-link = Terence Tao| last3 = Ziegler | first3 = Tamar | author3-link = Tamar Ziegler| mr = 2950773| arxiv = 1009.3998}}