Kneser’s type theorems in measure-preserving systems with applications to additive combinatorics

Abstract: 

The classical theorem of Kneser gives sufficient conditions on A, and B (subsets of the natural numbers)  which guarantee that lower asymptotic density of A+B = {a+b | a in A, b in B} is bigger or equal than the sum of lower asymptotic densities of A and B. Given a set A in the integers (Z), and a set B of positive measure in an ergodic  probability measure preserving Z-system (X,m) [the concept will be explained in the talk], it is natural to ask for a lower bound on the measure of the translates of B by elements of the set A, i.e.,  m(AB). For instance, if A contains arbitrary long intervals then ergodicity of the system implies that m(AB) = 1.

We will present new results in this direction, including the following one:

If A is not contained in any proper periodic set of the integers, then m(AB) >= min{1, d*(A) + m(B)}, where d*(A) is the supremum of all invariant densities of the set A.

From our dynamical results we obtain new Kneser’s type theorems for lower asymptotic densities. The talk is based on a joint work with M. Bjorklund (Chalmers).