A Refinement of Cauchy-Schwarz Complexity, with Applications

Abstract

Let $\varPhi :=(\phi _i)_{i\in \;I}$ be a finite collection of linear forms $\phi _i:\mathbb {F}_p^d\rightarrow \mathbb {F}_p$. We introduce a 2-parameter refinement of Cauchy-Schwarz (CS) complexity, called sequential Cauchy-Schwarz complexity. We prove that if $\varPhi$ has sequential Cauchy-Schwarz complexity at most $(k,\ell )$, then $|\mathbb {E}_{x_1,\ldots ,x_d\in \mathbb {F}_p^n}\prod _{i\in \;I}f_i(\phi _i(x_1,\ldots ,x_d))|\le \min _{i\in I}\Vert f_i\Vert _{U^{k+1}}^{2^{1-\ell }}$ for any 1-boun- ded functions $f_i:\mathbb {F}_p^n\rightarrow \mathbb {C}$, $i\in I$. For $\ell =1$, this reduces to CS complexity, but for larger $\ell$ the two notions differ. For example, let $S_{k,M}:=\{z\in [0,p-1]^M:z_1+\cdots +z_M<k\}$, and consider $\varPhi _{k,M}:=\big \{\phi _z(x,t_1,\ldots ,t_M):=x+z_1t_1+\cdots +z_Mt_M\;|\;z \in \;S_{k,M}\big \}$, a multivariable generalization of arithmetic progressions. We show that $\varPhi _{k,M}$ has sequential CS complexity at most $(\min (k,M(p-1)+1)-2,\ell )$ for some finite $\ell$, yet can have CS complexity strictly larger than $\min (k,M(p-1)+1)-2$. Moreover, we show that $\varPhi _{k,M}$ has True complexity $\min (k,M(p-1)+1)-2$.

In [2], we use these results in a new proof of the inverse theorem for $\mathbb {F}_p^n$.