Applications of additive combinatorics to Diophantine equations

The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the Brauer-Manin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over \(\mathbb Q\) when the degenerate fibres are all defined over \(\mathbb Q\). In this setting the result of Green, Tao and Ziegler replaces Schinzel's Hypothesis (H) used in earlier papers of Colliot-Thélène and Sansuc. I will give an overview of recent work in this direction due to L. Matthiesen, T. Browning, Y. Harpaz, O. Wittenberg and the speaker.