  Let $\Delta=\Delta_1\times\ldots\times \Delta_d\subseteq\mathbb{R}^n$, where
$\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each
$\Delta_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points.
  We prove that any set $S\subseteq \mathbb{R}^n$, with $n=n_1+\cdots +n_d$ of
positive upper Banach density necessarily contains an isometric copy of all
sufficiently large dilates of the configuration $\Delta$. In particular any
such set $S\subseteq \mathbb{R}^{2d}$ contains a $d$-dimensional cube of side
length $\lambda$, for all $\lambda\geq \lambda_0(S)$.
  We also prove analogous results with the underlying space being the integer
lattice. The proof is based on a weak hypergraph regularity lemma and an
associated counting lemma developed in the context of Euclidean spaces and the
integer lattice.
