Good Bounds in Certain Systems of True Complexity One
Abstract
Let $\Phi = (\phi_1,\dots,\phi_6)$ be a system of $6$ linear forms in $3$ variables, i.e. $\phi_i \colon \mathbb{Z}^3 \to \mathbb{Z}$ for each $i$. Suppose also that $\Phi$ has CauchySchwarz 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 $f_1,\dots,f_6 \colon G \to \mathbb{C}$ are functions with $\f_i\_{\infty} \le 1$ for each $i$, then for each $j$, $1 \le j \le 6$: \[ \left \mathbb{E}_{x_1,x_2,x_3 \in G} f_1(\varphi_1(x_1,x_2,x_3)) \dots f_6(\phi_6(x_1,x_2,x_3)) \right \le \f_j\_{U^2}^{1/C} \] 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 CauchySchwarz 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 DOI:
 10.48550/arXiv.1705.06801
 arXiv:
 arXiv:1705.06801
 Bibcode:
 2017arXiv170506801M
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics
 EPrint:
 40 pages