Geometric mean theorem

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my page - www.hostgator1centcoupon.info 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

$\Vert f{\Vert }_{U^d(g)}^{2^d}={\mathbf{E}}_{x,h_1,\dots ,h_d\in G}\prod _{{\omega }_1,\dots ,{\omega }_d\in \{0,1\}}{J}^{{\omega }_1+\cdots +{\omega }_d}f\left(x+h_1{\omega }_1+\cdots +h_d{\omega }_d\right).$

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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