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WP:CHECKWIKI errors fix. Do general fixes if a problem exists

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{{redirect|Uniformity norm|the function field norm|uniform norm|unformity in topology|uniform space}}
In mathematics, in the field of [[additive combinatorics]], a '''Gowers norm''' or '''uniformity norm''' is a class of [[Norm (mathematics)|norm]] on functions on a finite [[Group (mathematics)|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]].

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

:<math> \Vert f \Vert_{U^d(g)}^{2^d} = \mathbf{E}_{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) \ . </math>

The ''inverse conjecture'' for these norms is the statement that if ''f'' has [[L-infinity norm]] ([[uniform norm]] in the usual sense) equal to 1 then the Gowers ''s''-norm is bounded above by 1, with equality if and only if ''f'' is of the form exp(2πi ''g'') with ''g'' a polynomial of degree at most ''s''. This can be interpreted as saying that the Gowers norm is controlled by polynomial phases.

The inverse conjecture holds for vector spaces over a finite field. However, for cyclic groups '''Z'''/''N'' this is not so, and the class of polynomial phases has to be extended to control the norm.

==References==
* {{cite book | zbl=pre06110460 | last=Tao | first=Terence | authorlink=Terence Tao | title=Higher order Fourier analysis | series=Graduate Studies in Mathematics | volume=142 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2012 | isbn=978-0-8218-8986-2 | url=http://terrytao.wordpress.com/books/higher-order-fourier-analysis/ }}

[[Category:Additive combinatorics]]

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