Rational points on pencils of conics and quadrics with many degenerate fibres

For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer-Manin obs truction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics overQ, which is a consequence of recent advances in additive combinatorics.