Good Bounds in Certain Systems of True Complexity One

Abstract

Let $\Phi =({\varphi }_1,\dots ,{\varphi }_6)$ be a system of $6$ linear forms in $3$ variables, i.e. ${\varphi }_i:{\mathbb{Z}}^3\to \mathbb{Z}$ for each $i$. Suppose also that $\Phi$ has Cauchy--Schwarz complexity $2$ and true complexity $1$, in the sense defined by Gowers and Wolf; in fact this is true generically in this setting. Finally let $G={\mathbb{F}}_p^n$ for any $p$ prime and $n\ge 1$. Then we show that multilinear averages by $\Phi$ are controlled by the $U^2$-norm, with a polynomial dependence; i.e. if $f_1,\dots ,f_6:G\to \mathbb{C}$ are functions with $\Vert f_i{\Vert }_{\infty }\le 1$ for each $i$, then for each $j$, $1\le j\le 6$: \[ \left| \mathbb{E}_{x_1,x_2,x_3 \in G} f_1(φ_1(x_1,x_2,x_3)) \dots f_6(ϕ_6(x_1,x_2,x_3)) \right| \le \|f_j\|_{U^2}^{1/C} \] for some $C>0$ depending on $\Phi$. This recovers and strengthens a result of Gowers and Wolf in these cases. Moreover, the proof uses only multiple applications of the Cauchy--Schwarz inequality, avoiding appeals to the inverse theory of the Gowers norms. We also show that some dependence of $C$ on $\Phi$ is necessary; that is, the constant $C$ can unavoidably become large as the coefficients of $\Phi$ grow.

40 pages

Cited by in corpus (2)

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