Ergodic Ramsey Theory at the Junction of Additive and Multiplicative Combinatorics
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). For example, multiplicatively large sets in N contain arbitrarily long geometric progressions. One can show, that, somewhat surprisingly, multiplicatively large sets contain also arbitrarily long arithmetic progressions. Some recent developments related to Sarnak's Mobius Disjointness Conjecture reveal new interesting connections between the theory of multiple recurrence and multiplicative number theory. The goal of this talk is to discuss some known results, open problems and conjectures pertaining to the interaction of the additive and multiplicative structures in N.