Control and its applications in additive combinatorics

Abstract: We prove new quantitative bounds on the additive structure of sets obeying an $L<sup>3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growth of convex sets.