Multidimensional Szemerédi's Theorem

Statement

The multidimensional Szemerédi theorem states that for any fixed finite subset $S\subseteq\mathbb{Z}^d$, and for every $\varepsilon>0$, there exists a positive integer $N(S,\varepsilon)$ such that for any $N\ge N(S,\varepsilon)$, any subset $A\subseteq\{1,\ldots,N\}^d$ with size at least $\varepsilon N^d$ contains a subset of the form $a\cdot S+h$. This theorem follows from the multidimensional corners theorem by a simple projection argument. In particular, Roth's theorem on arithmetic progressions follows directly from the ordinary corners theorem.