By analogy with the classical notions of density in the set N of natural numbers, one can introduce notions of density which are geared towards the multiplicative structure of N. Various combinatorial results involving additively large sets in (N,+) (such as, for instance, Szemeredi's theorem on arithmetic progressions and its polynomial extensions) have natural analogs in the multiplicative semigroup (N,x). As we will see, the methods of ergodic theory are well suited to tackle the problems which naturally arise in this context. In particular, we will show that multiplicatively large sets have a very rich combinatorial structure, both multiplicative and (somewhat surprisingly) additive.
If time permits, we will also discuss some recent developments related to Sarnak's Mobius Disjointness Conjecture which reveal new interesting connections between the theory of multiple recurrence and multiplicative number theory. We will conclude with discussing some natural open problems and conjectures pertaining to the interaction of the additive and multiplicative structures in N.