Gowers norm

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.^[2]

Definition

Let ${\displaystyle f}$ be a complex-valued function on a finite abelian group ${\displaystyle G}$ and let ${\displaystyle J}$ denote complex conjugation. The Gowers ${\displaystyle d}$-norm is

${\displaystyle \Vert f\Vert _{U^{d}(G)}^{2^{d}}=\sum _{x,h_{1},\ldots ,h_{d}\in G}\prod _{\omega _{1},\ldots ,\omega _{d}\in \{0,1\}}J^{\omega _{1}+\cdots +\omega _{d}}f\left({x+h_{1}\omega _{1}+\cdots +h_{d}\omega _{d}}\right)\ .}$

Gowers norms are also defined for complex-valued functions f on a segment ${\displaystyle [N]={0,1,2,...,N-1}}$, where N is a positive integer. In this context, the uniformity norm is given as ${\displaystyle \Vert f\Vert _{U^{d}[N]}=\Vert {\tilde {f}}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}/\Vert 1_{[N]}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}}$, where ${\displaystyle {\tilde {N}}}$ is a large integer, ${\displaystyle 1_{[N]}}$ denotes the indicator function of [N], and ${\displaystyle {\tilde {f}}(x)}$ is equal to ${\displaystyle f(x)}$ for ${\displaystyle x\in [N]}$ and ${\displaystyle 0}$ for all other ${\displaystyle x}$. This definition does not depend on ${\displaystyle {\tilde {N}}}$, as long as ${\displaystyle {\tilde {N}}>2^{d}N}$.

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 ${\displaystyle \mathbb {F} }$ asserts that for any ${\displaystyle \delta >0}$ there exists a constant ${\displaystyle c>0}$ such that for any finite-dimensional vector space V over ${\displaystyle \mathbb {F} }$ and any complex-valued function ${\displaystyle f}$ on ${\displaystyle V}$, bounded by 1, such that ${\displaystyle \Vert f\Vert _{U^{d}[V]}\geq \delta }$, there exists a polynomial sequence ${\displaystyle P\colon V\to \mathbb {R} /\mathbb {Z} }$ such that

${\displaystyle \left|{\frac {1}{|V|}}\sum _{x\in V}f(x)e\left(-P(x)\right)\right|\geq c,}$

where ${\displaystyle e(x):=e^{2\pi ix}}$. This conjecture was proved to be true by Bergelson, Tao, and Ziegler.^[3][4][5]

The Inverse Conjecture for Gowers ${\displaystyle U^{d}[N]}$ norm asserts that for any ${\displaystyle \delta >0}$, a finite collection of (d − 1)-step nilmanifolds ${\displaystyle {\mathcal {M}}_{\delta }}$ and constants ${\displaystyle c,C}$ can be found, so that the following is true. If ${\displaystyle N}$ is a positive integer and ${\displaystyle f\colon [N]\to \mathbb {C} }$ is bounded in absolute value by 1 and ${\displaystyle \Vert f\Vert _{U^{d}[N]}\geq \delta }$, then there exists a nilmanifold ${\displaystyle G/\Gamma \in {\mathcal {M}}_{\delta }}$ and a nilsequence ${\displaystyle F(g^{n}x)}$ where ${\displaystyle g\in G,\ x\in G/\Gamma }$ and ${\displaystyle F\colon G/\Gamma \to \mathbb {C} }$ bounded by 1 in absolute value and with Lipschitz constant bounded by ${\displaystyle C}$ such that:

${\displaystyle \left|{\frac {1}{N}}\sum _{n=0}^{N-1}f(n){\overline {F(g^{n}x}})\right|\geq c.}$

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.

References

• Tao, Terence (2012). Higher order Fourier analysis. Graduate Studies in Mathematics. Vol. 142. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8986-2. MR 2931680. Zbl 1277.11010.