Khintchine's Theorem for Symmetric matrices via Flows on the Space of Symplectic Lattices

Abstract

We establish Diophantine approximation results for real symmetric matrices by collections of linearly independent integer vectors. For $X\in {\mathrm{Sym}}_d(\mathbb{R})$, we prove a Dirichlet-type theorem guaranteeing the existence of integral Lagrangian frames $(Q,P)\in {\mathrm{Mat}}_{d\times 2d}(\mathbb{Z})$ that satisfy $\Vert QX+P{\Vert }_{\mathrm{op}}\le c_d/N$ and $\Vert Q{\Vert }_{\mathrm{op}}\le N$ for any $N\ge 1$. Furthermore, we establish a Khintchine-type zero-one law, demonstrating that the size of the set of $\psi$-approximable symmetric matrices is determined by the convergence or divergence of the series ${\sum }_{q\ge 1}{q}^{\varsigma -1}\psi (q{)}^{\varsigma }$, where $\varsigma =d(d+1)/2$. The proofs rely on the reduction theory of the Siegel upper half-space, dynamical formulation over the space of symplectic lattices, and an analysis of the Siegel transform adapted to count Lagrangian frames instead of single lattice points.

19 pages