  We prove that any given subset of $\mathbb{Z}^d$ of upper density $\delta>0$
will necessarily contain, in an appropriate sense depending on $\delta$, an
isometric copy of all large dilates of any given non-degenerate $k$-simplex,
provided $d\geq 2k+3$. This provides an improvement in dimension, from $d\geq
2k+5$, on earlier work of Magyar. We in fact establish a stronger pinned
variant. Key to our approach are new $\ell^2$ estimates for certain discrete
multilinear maximal operators associated to simplices. These operators are
generalizations of the discrete spherical maximal operator and may be of
independent interest.
