Simultaneous popular polynomial differences in higher dimensions

Source 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 $P_1,\ldots,P_k\in\mathbb{Z}[t]$ be linearly independent polynomials with zero constant terms, and let $v_1,\ldots,v_k\in\mathbb{Z}^D\setminus\{0\}$. We show that, for every $\varepsilon>0$, every sufficiently large prime $p$, and every $A\subseteq\mathbb{F}_p^D$ of density $α$, 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 $ω\in\{0,1\}^k$. The main new ingredient is a directional arithmetic regularity decomposition adapted to mixed directional box norms.