Consider a set A of reals such that every small subset of A spans many differences (that is, every small subset has a large difference set). What can we say about the size of the difference set A-A? We wish to rely on local properties of small subsets of A to conclude a global property of A.
We will present several recent results for the above local properties problem.
This progress was made by studying properties of the additive energy.
We will also see how such energy techniques lead to progress on graph theoretic problems, even though these problems involve no algebra.
Joint work with Sara Fish, Ben Lund, and Cosmin Pohoata.