Multiplicative richness of additively large sets in Zd

In their proof of the IP Szemerédi theorem, a far reaching extension of the classic theorem of Szemerédi on arithmetic progressions, Furstenberg and Katznelson [14] introduced an important class of additively large sets called IPr⁎ sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results. The authors recently showed that additive IPr⁎ subsets of Zd are multiplicatively rich with respect to every multiplication on Zd 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 Zd. We show, for example, that in contrast to the case for Z, there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of Zd when d≥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 IPr⁎ sets are multiplicatively rich in infinitely many genuinely different ways. We conclude by cataloging a number of sources of additive IPr⁎ sets from combinatorics and dynamics.