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.
be a complex-valued function on a finite abelian group
denote complex conjugation.
-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
is a large integer,
denotes the indicator function of [N], and
This definition does not depend on
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
there exists a constant
such that for any finite-dimensional vector space V over
and any complex-valued function
, there exists a polynomial sequence
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
can be found, so that the following is true.
is a positive integer and
is bounded in absolute value by 1 and
, then there exists a nilmanifold
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.