Lagrangian Frame Approximation: From Dirichlet to Khintchine

Classical Diophantine approximation asks for a single integer vector giving a good approximation. Requiring instead a full frame of linearly independent vectors generally leads to a loss in the Dirichlet exponent. In this talk, I will show that for symmetric matrices this loss disappears when the approximating frames are required to be Lagrangian: one recovers the classical exponent . I will also discuss a Khintchine-type theorem for approximation by primitive Lagrangian frames. The proofs are based on reduction theory and homogeneous dynamics on the space of symplectic lattices, together with a frame-counting version of the Siegel transform.