Gowers norm

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"Uniformity norm" redirects here. For the function field norm, see uniform norm. For unformity in topology, see uniform space.

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.

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

 \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) \ .

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.

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