Applications of additive combinatorics to conic and quadric bundles

Methods of Green and Tao can be used to prove the Hasse principle and weak approximation for some special intersections of quadrics defined over $Q$ (arithmetic progressions in primes are replaced by arithmetic progressions in integers represented by binary quadratic forms).
This implies that the Brauer-Manin obstruction controls weak approximation on conic bundles with an arbitrary number of degenerate fibres, all defined over $Q$, and some similar varieties. All previous results were restricted to conic bundle surfaces with a small number of degenerate fibres (up to 6).
This is a joint work with Tim Browning and Lilian Matthiesen.