A multidimensional Szemer\'{e}di theorem in integers

TL;DR

This paper proves a multidimensional Szemerédi theorem for integer grids, showing that subsets with sufficient density contain specific polynomial configurations, extending previous two-dimensional results.

Contribution

It generalizes a recent two-dimensional result to higher dimensions, establishing a quantitative density condition for polynomial configurations in integer grids.

Findings

Subsets with density at least (log N)^{-c} contain the specified configurations.

The theorem extends previous 2D results to n-dimensional settings.

An effective 'popular' version of the theorem is developed.

Abstract

For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ 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 $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemer\'{e}di theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version.