Simultaneous popular polynomial differences in higher dimensions

Abstract

We extend the simultaneous popular polynomial difference theorem of Conlon, Dong, and the author to polynomial shifts along prescribed directions in ${\mathbb{F}}_p^D$. Let $P_1,\dots ,P_k\in \mathbb{Z}[t]$ be linearly independent polynomials with zero constant terms, and let $v_1,\dots ,v_k\in {\mathbb{Z}}^D\setminus \{0\}$. We show that, for every $\epsilon >0$, every sufficiently large prime $p$, and every $A\subseteq {\mathbb{F}}_p^D$ of density $\alpha$, there exists a nonzero $d\in {\mathbb{F}}_p$ such that \[ \mathbb{E}_{\underline{x}\in\mathbb{F}_p^D} \mathbf{1}_A(\underline{x}) \prod_{\substack{1\leq i\leq k\\ ω_i=1}} \mathbf{1}_A\bigl(\underline{x}+P_i(d)v_i\bigr) \geq α^{\,1+\sum_{i=1}^kω_i}-\varepsilon \] simultaneously for every $\omega \in \{0,1{\}}^k$. The main new ingredient is a directional arithmetic regularity decomposition adapted to mixed directional box norms.

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