Special Year Learning Seminar

The "intersectivity lemma" states that if a ∈ (0,1) and A_n, n ∈ N,  are measurable sets in a probability space (X,m) satisfying  m(A_n) ≥ a for all n, then there exist a subsequence n_k, k ∈ N, which has positive upper density and such that the intersection of any finite subfamily of A_n along (n_k) has positive measure. After presenting a (short) proof of this lemma, we will discuss some applications. Here is a sample:

• If E is a set of positive upper density in Z^2  then there exists a set B in Z  which has positive upper density and such that the set of differences 

E - E  = {x - y : x, y ∈ E} contains the Cartesian product B x B.

• If E is a set of natural numbers which is multiplicatively large, then E contains arbitrarily long arithmetic progressions (and arbitrarily long geometric progressions).