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  "body": "\n## Statement\n\nThe multidimensional Szemerédi theorem states that for any fixed finite subset $S\\subseteq\\mathbb{Z}^d$, and for every $\\varepsilon\u003e0$, 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.\n",
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