Furstenberg-Katznelson theorem ($a+tF\subseteq A$)

The multidimensional Szemerédi theorem says that a subset $A\subseteq {\mathbb{Z}}^d$ with positive upper density contains a translated positive integer dilate $a+tF$ of every finite set $F\subseteq {\mathbb{Z}}^d$. In the finite formulation, for every $\alpha >0$ and finite $F$, all sufficiently large integer boxes have this property for every subset occupying at least an $\alpha$ fraction of the box. The one-dimensional case for $F=\{0,1,\dots ,r-1\}$ is the Szemerédi theorem. The general result follows from multiple recurrence for commuting measure-preserving transformations.