Multiplicative richness of additively large sets in $\mathbb{Z}^d$

Abstract: 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.