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.
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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
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
where
The Inverse Conjecture for Gowers
This conjecture was proved to be true by Green, Tao, and Ziegler. 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.