An Approximate Counting Version of the Multidimensional Szemerédi Theorem

For any fixed $d\ge 1$ and subset X of $\mathbb {N}^d$, let $r_X(n)$ be the maximum cardinality of a subset A of $\{1,\dots,n\}^d$ which does not contain a subset of the form $\varvec{b} + rX$ for $r>0$ and $\varvec{b} \in \mathbb {R}^d$. Such a set A is said to be X-free. The Multidimensional Szemerédi Theorem of Furstenberg and Katznelson states that $r_X(n)=o(n^d)$. We show that, for $|X|\ge 3$ and infinitely many $n\in \mathbb {N}$, the number of X-free subsets of $\{1,\dots,n\}^d$ is at most $2^{O(r_X(n))}$. The proof involves using a known multidimensional extension of Behrend’s construction to obtain a supersaturation theorem for copies of X in dense subsets of $[n]^d$ for infinitely many values of n and then applying the powerful hypergraph container lemma. Our result generalizes work of Balogh, Liu, and Sharifzadeh on k-AP-free sets and Kim on corner-free sets.