A refinement of Cauchy-Schwarz complexity

We introduce a notion of complexity for systems of linear forms called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers $k,ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-ell}$-th power of the Gowers $U^{k+1}$-norms of the functions. For $ell=1$ this agrees with Cauchy-Schwarz complexity, but for $ell>1$ there are families of systems that have sequential Cauchy-Schwarz complexity at most $(k,ell)$ whereas their Cauchy-Schwarz complexity is greater than $k$. For instance, for $p$ prime and $kin mathbb{N}$, the system of forms $big{phi_{z_1,z_2}(x,t_1,t_2)= x+z_1 t_1+z_2t_2;|; z_1,z_2in [0,p-1], z_1+z_2<kbig}$ can be viewed as a $2$-dimensional analogue of arithmetic progressions of length $k$. We prove that this system has sequential Cauchy-Schwarz complexity at most $(k-2,ell)$ for some $ell=O_{k,p}(1)$, even for $p<k$, whereas its Cauchy-Schwarz complexity can be strictly greater than $k-2$. In fact we prove this for the $M$-dimensional analogues of these systems for any $Mgeq 2$, obtaining polynomial true-complexity bounds for these and other families of systems. In a separate paper, we use these results to give a new proof of the inverse theorem for Gowers norms on vector spaces $mathbb{F}_p^n$, and applications concerning ergodic actions of $mathbb{F}_p^{omega}$.