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For $f:G\to M_n(\mathbb{C})$ on a finite group, use uniform expectation over the solutions of $x{y}^{-1}z{w}^{-1}=e$ and the ordinary, unnormalized trace to define the displayed fourth-order quantity. If $T_{\rho }$ is its matrix Fourier block, this quantity equals ${\sum }_{\rho }{d}_{\rho }\mathrm{Tr}((T_{\rho }{T}_{\rho }^{\ast }{)}^2)$, so it is real and nonnegative. For a scalar function on an abelian group, it reduces to the usual fourth power of the Gowers uniformity norm of order two. The unnormalized trace means a unitary matrix-valued function can have fourth-order quantity as large as $n$.

A product that cancels constant phases and lowers the degree of a polynomial phase. For $f(x)=e(\alpha x^2)$, it is $e(2\alpha hx+\alpha h^2)$. Some conventions conjugate the entire expression and reverse the frequency signs. The Gowers uniformity norm derivative identity relates a higher norm to the average of lower norms of these derivatives.

We use the normalized Gowers U3 norm on an interval, for which a quadratic phase has norm one. Let ${\mathcal{C}}_N$ consist of integer tuples $(x,h_1,h_2,h_3)$ whose eight vertices $x+{\omega }_1{h}_1+{\omega }_2{h}_2+{\omega }_3{h}_3$, $\omega \in \{0,1{\}}^3$, all lie in $[1,N]$. If $\mathcal{C}$ denotes complex conjugation, defineEquivalently, choose a prime $8N<M<16N$, extend $f$ by zero to $G=\mathbb{Z}/M\mathbb{Z}$, call that extension $F$, and put $I=1_{[1,N]}$. ThenThe large ambient modulus prevents wraparound in cubes supported on the interval, so the ratio is independent of the chosen such $M$. This also proves nonnegativity and the norm properties by the Gowers uniformity norm on $G$. Some conventions omit the denominator; their interval norm differs by a fixed bounded factor, and the quadratic-phase norm is then $\Vert I{\Vert }_{U^3(G)}$.

For $f_0(x)=e(\alpha x^2)$, the exponent in every conjugated cube product is a third additive difference of a quadratic polynomial and is zero. Thus $\Vert f_0{\Vert }_{U^3(N)}=1$ in the normalized interval convention.

The generalized von Neumann inequality for four-term progressions connects this norm to counting arithmetic progressions. For functions bounded by one on a cyclic group of prime order greater than three,Three applications of Cauchy-Schwarz prove the bound. Zero extension and division by the number of interval progressions give the analogous interval estimate up to an absolute constant. In particular, if $g=1_A-\alpha 1_{[1,N]}$, where $\alpha =|A|/N$, has small Gowers U3 norm on an interval, expansion of the progression count shows that it differs from ${\alpha }^4$ times the count for $[1,N]$ by $O(\Vert g{\Vert }_{U^3(N)}{N}^2)$. A large Gowers U3 norm detects structure capable of changing four-term progression counts; quadratic phases are the basic example.

For the remaining proof use the ambient $G$ just specified, define ${\partial }_{h}F(x)=F(x+h)\overline{F(x)}$, and take averages uniformly on $G$. For frequencies not on the character grid, an exponential is evaluated at the chosen integer representatives; the general proof below selects actual characters $\theta =k/M$. The printed question does not define $G$ or the interval normalization, so these conventions make the assertion precise. If instead ${\partial }_{h}f(x)=f(x)\overline{f(x+h)}$ is used, conjugate the correlations and reverse every frequency sign; the quadratic example then has $\theta (h)=-2\alpha h$.

The Fourier density-increment proof of the Meshulam theorem controls three-term progressions through ordinary Fourier coefficients, equivalently the $U^2$ norm. Four-term progressions are controlled by the Gowers uniformity norm $U^3$. A function can have small correlation with every linear character while correlating strongly with a quadratic phase; such a function can have small $U^2$ norm but large $U^3$ norm. Ordinary Fourier uniformity therefore does not make the four-term progression count pseudorandom, and a large linear Fourier coefficient need not exist when that count is deficient. Quadratic or higher-order Fourier structure is needed.

For functions $(f_{ϵ}{)}_{ϵ\in \{0,1{\}}^3}$ on ${\mathbb{F}}_p^n$, the Gowers inner product iswhere $\mathcal{C}$ is the complex conjugate operator. The Gowers uniformity norm isThe Gowers-Cauchy-Schwarz inequality states