On weighted multilinear polynomial averages in finite fields

Abstract

We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Teräväinen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of ${\mathbb{F}}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(φ(y))v_1,\cdots, \textbf{x}+ P_k(φ(y))v_k, \] where $\mathbb{V}=\{v_1,\cdots ,v_k\in {\mathbb{Z}}^D\}$ is a collection of nonzero vectors, $\mathbb{P}=\{P_1,\cdots ,P_k\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $\ddot{I}†(y)\in \mathbb{Q}(y)$ is a nonzero rational function.

11 pages