Local distribution of rational points in flag varieties

The Manin–Peyre principle predicts that rational points of bounded height (properly normalised) on Fano-type varieties are equidistributed in the adelic space, and in particular in the real locus. A local version of this principle is concerned with how rational points are distributed "around" a fixed real point. The Diophantine approximation property of the given point plays a key role. We shall present various results for quadrics based on the Hardy--Littlewood circle method (joint work with D. Schindler and A. Shute), and for general flag varieties based on homogeneous dynamics (joint work with N. de Saxcé).