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 , 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 functionf 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.[3][4][5]
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.[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.