A multidimensional Szemerédi theorem in integers

For any integer $n\ge 2$, let $(m_1,\dots ,m_n)$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N{]}^n$ of density at least $(\log N{)}^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_1,\dots ,m_n)$ is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),(x,y+r^2)$. The theorem is obtained as a consequence of an effective ``popular’’ version.