Applications of additive combinatorics to rational points.

Abstract
Spectacular results in additive combinatorics provide a powerful tool to prove existence and density statements for rational points on certain algebraic varieties over Q. I will give a survey of recent work by Browning, Matthiesen, the speaker, and a subsequent work by Harpaz and Wittenberg leading to a result about families of rationally connected varieties parameterised by the projective line: if the Brauer-Manin obstruction controls rational points in the smooth fibres, then it controls rational points on the total space.