  In their proof of the IP Szemer\'edi theorem, a far reaching extension of the
classic theorem of Szemer\'edi on arithmetic progressions, Furstenberg and
Katznelson introduced an important class of additively large sets called
$\text{IP}_{\text{r}}^*$ sets which underlies recurrence aspects in dynamics
and is instrumental to enhanced formulations of combinatorial results. The
authors recently showed that additive $\text{IP}_{\text{r}}^*$ subsets of
$\mathbb{Z}^d$ are multiplicatively rich with respect to every multiplication
on $\mathbb{Z}^d$ without zero divisors (e.g. multiplications induced by degree
$d$ number fields). In this paper, we explain the relationships between classes
of multiplicative largeness with respect to different multiplications on
$\mathbb{Z}^d$. We show, for example, that in contrast to the case for
$\mathbb{Z}$, there are infinitely many different notions of multiplicative
piecewise syndeticity for subsets of $\mathbb{Z}^d$ when $d \geq 2$. This is
accomplished by using the associated algebra representations to prove the
existence of sets which are large with respect to some multiplications while
small with respect to others. In the process, we give necessary and sufficient
conditions for a linear transformation to preserve a class of multiplicatively
large sets. One consequence of our results is that additive
$\text{IP}_{\text{r}}^*$ sets are multiplicatively rich in infinitely many
genuinely different ways. We conclude by cataloging a number of sources of
additive $\text{IP}_{\text{r}}^*$ sets from combinatorics and dynamics.
