Product of simplices and sets of positive upper density in $\mathbb{R}^d$

We establish that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\geq4$. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if $\Delta_{k_1}$ and $\Delta_{k_2}$ are two fixed non-degenerate simplices of $k_1+1$ and $k_2+1$ points respectively, then any subset of $\mathbb{R}^d$ of positive upper Banach density with $d\geq k_1+k_2+6$ will necessarily contain an isometric copy of all sufficiently large dilates of $\Delta_{k_1}\times\Delta_{k_2}$. A new direct proof of the fact that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of $k+1$ points provided $d\geq k+1$, a result originally due to Bourgain, is also presented.
Comment: Substantial revisions made. To appear in Math. Proc. Cambridge Philos. Soc