Diophantine Approximation by Lagrangian Frames in Symplectic Lattices

Abstract

Classical Diophantine approximation searches for a short integral vector. In higher rank, one may retain either the rational subspace spanned by several such vectors or the integral frame itself. We study the latter problem for primitive Lagrangian frames $(Q,P)\in {\mathrm{Mat}}_{d\times 2d}(\mathbb{Z})$ and symmetric matrices. We first prove a Dirichlet theorem showing that, for symmetric targets, primitive Lagrangian frames attain the classical approximation rate: for every $X\in {\mathrm{Sym}}_d(\mathbb{R})$ and $N\ge 1$, there exists such a frame satisfying $\Vert QX+P{\Vert }_{\mathrm{op}}\le c_d/N$ and $\Vert Q{\Vert }_{\mathrm{op}}\le N$. This contrasts with unrestricted full frame approximation, where the optimal universal exponent is $1/d$. We also prove a Khintchine-type zero-one law with critical sum ${\sum }_{q\ge 1}{q}^{\varsigma -1}\psi (q{)}^{\varsigma }$, $\varsigma =\frac{d(d+1)}{2}$. Retaining the frame, rather than only its span, creates an infinite change-of-basis multiplicity: the naive frame-counting Siegel transform has infinite second moment for $d\ge 3$. We construct a conical regularization of the frame space which reduces this multiplicity to a finite group while retaining a positive proportion at every scale. Combined with span separation, this gives a finite-overlap $L^2$ estimate and the required cusp-volume bounds. The regularization is formulated independently of the symplectic condition.

arXiv comment27 pages