We say that a collection of integer vectors $u_1,\ldots,u_\ell \in \mathbb{Z}^d$ has Cauchy-Schwarz complexity $\leq t$ with denominator $Q$ at position $1\leq j\leq \ell$ if there exist $\sigma_1,\ldots,\sigma_{t+1}\in \mathbb{Z}^d$ such that

- $u_j\cdot \sigma_r\neq 0$ and divides $Q$ for each $1\leq r\leq t+1$,</ul>
- for all $i\neq j$ there exists some $1\leq r\leq t+1$ such that $u_i\cdot \sigma_r=0$.

A simple example is the collection of vectors that determine a $k$-term arithmetic progression, namely $u_1,\ldots,u_k = (1,0),\ldots,(1,k-1)$. This determines an arithmetic progression in that as $(x,d)$ ranges over $H^2$, for some abelian group $H$, the set $\{ u_i\cdot (x,d)\}$ ranges over the arithmetic progressions of length $k$. This collection has Cauchy-Schwarz complexity $k-1$, and hence the behaviour of arithmetic progressions of length $k$ is determined by the $U^{k-1}$ norm.

A more elaborate example, which is that used by Manners in his proof of Lemma 3.4.5 of [Ma], is the collection of vectors in $\mathbb{Z}^{k+2}$ given by \[u_\eta = (1,\eta,0)\textrm{ for }\eta\subset [k]\textrm{ and }\] \[v_\nu = (1,\nu,1)\textrm{ for }\nu\subset[k]\textrm{ with }i\in \nu\textrm{ and }\Abs{\nu}\leq R.\] This has Cauchy-Schwarz complexity at most $R-1$ with denominator $(k-1)!$ at any $\nu_0\subset [k]$ with $i\in \nu_0$ and size $\Abs{\nu_0}\leq R$ (note in particular that the complexity is much smaller than the size). Indeed, we can take the vectors $(0,\ldots,0,1)$, and $(0,\ldots,1,\ldots,0)$ which is only non-zero at some fixed $j\neq i\in \nu_0$, and for $\Abs{\nu_0}\leq r\leq R-1$ we take the vector $(-r-2,1,\ldots,1)$. This is a collection of $R$ vectors, which is easily verified to have the required properties.

The use of Cauchy-Schwarz complexity is that it allows us to convert information about functions 'along a collection of linear forms' into quite generic additive information about those functions. We define the discrete derivative $\partial_h f(x) = f(x+h)-f(x)$, and extend this to vectors in the obvious fashion. One can characterise polynomials of degree $s$ (when, say, dealing with functions $\mathbb{R}\to\mathbb{R}$) by saying that the derivatives of order $s+1$ vanish. Inspired by this, we say that a function $f:H\to \mathbb{R}$ is an $\epsilon$-approximate polynomial of degree $s$ if there are at least $\epsilon \lvert H\rvert^{s+2}$ many $(x,h_1,\ldots,h_{s+1})\in H^{s+2}$ such that $\partial_{h_1,\ldots,h_{s+1}}f(x)=0$. The following lemma (which is basically Lemma 3.5.3 of [Ma]) shows that functions which correlate along systems of low complexity must be approximate polynomials.

Suppose that $u_1,\ldots,u_\ell\in \mathbb{Z}^d$ are a collection of integer vectors of Cauchy-Schwarz complexity $\leq t$ with denominator $Q$ at position $j$, and suppose that $f_1,\ldots,f_\ell:H\to\bbr$ are such that
\[\Abs{\left\{ x\in H^d : \sum_i f_i(u_i\cdot x)=0\right\}}\geq \delta\Abs{H}^d.\]
Then, provided $(\Abs{H},Q)=1$, the function $f_j$ is a $\delta^{2^{t+1}}$-approximate polynomial of degree $t$.

Note that the conclusion is precisely saying that $\sum (-1)^{\lvert \omega\rvert} f_j(\omega\cdot (x,h_1,\ldots,h_{t+1}))=0$ vanishes for many $(x,h_1,\ldots,h_{t+1})\in H^{t+2}$, where $\omega$ ranges over the vectors of the form $(1,\omega')$ as $\omega'\in\{0,1\}^{t+1}$ is arbitrary. This is a system of complexity $\leq t$ at any place. Thus one can view this lemma as moving from vanishing of a collection of functions along a system of complexity $\leq t$ to vanishing of a single one of those functions along another (more generic) system of complexity $\leq t$. Both aspects of this, the emphasis on one particular function of interest, and the genericity of the new system, are useful.
By construction, when $R=t+1$, the only remaining index is $j$, and so we have shown \[\sum_{x\in H^d} \sum_{\mathbf{h}\in H^{t+1}}1_{\partial_{\sigma_1h_1,\ldots,\sigma_{t+1}h_{t+1}}F_j(x)=0}\geq \delta^{2^{t+1}} \Abs{H}^{d+t+1}.\] Recalling that $F_j(x)=f_j(u_j\cdot x)$, we can write the left-hand side as \[\sum_{x'\in H}\sum_{\mathbf{h}'\in H^{t+1}}\sum_{\substack{x\in H^d\\ u_j\cdot x=x'}} \sum_{\substack{\mathbf{h}\in H^{t+1}\\ h_i(\sigma_i\cdot u_j)=h_i'}}1_{\partial_{\mathbf{h}'}f_j(x')=0},\] Since each $\sigma_i\cdot u_j$ is coprime to $\Abs{H}$, the sum over $\mathbf{h}$ is uniquely determined, and the sum over $x$ is at most $\Abs{H}^{d-1}$, and hence \[\sum_{x'\in H}\sum_{\mathbf{h}'\in H^{t+1}}1_{\partial_{\mathbf{h}'}f_j(x')=0},\geq \delta^{2^{t+1}} \Abs{H}^{t+2}\] as required.

**[GoWo]**W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. Lond. Math. Soc. (3) 100 (2010), no. 1, 155–176.**[Ma]**F. Manners, Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups. arXiv:1811.00718.