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 $\lVert QX + P \rVert_{\mathrm{op}} \leq c_d/N$ and $\lVert Q \rVert_{\mathrm{op}} \leq N$ for any $N \geq 1$. 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 $\sum_{q \geq 1} q^{\varsigma - 1}\psi(q)^{\varsigma}$, 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.