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) ∈ Matd × 2d(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∈Symd(ℝ) and N≥1, there exists such a frame satisfying lVert QX+PrVertop ≤ cd/N and lVert QrVertop ≤ 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 ∑q≥1qς−1ψ(q)ς, ς= (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 ≥ 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 L2 estimate and the required cusp-volume bounds. The regularization is formulated independently of the symplectic condition.
Related
- Metric Diophantine approximation on planar and Veronese curves
- Constructing Lagrangian surfaces in all symplectic toric $4$-manifolds from multi-grid diagrams
- Extreme value laws for intrinsic Diophantine approximation on spheres
- Diophantine approximation by primes and Landau--Siegel zeros
- A2-frames, negacyclic glue, and extremal strongly $6$-modular lattices
BibTeX
@misc{kim2026diophantine,
title = {{Diophantine Approximation by Lagrangian Frames in Symplectic Lattices}},
author = {Minchang Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.05096},
eprint = {2606.05096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.05096},
}
RIS
TY - UNPB
TI - Diophantine Approximation by Lagrangian Frames in Symplectic Lattices
AU - Minchang Kim
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2606.05096
UR - https://arxiv.org/abs/2606.05096
AB - 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) ∈ Mat_d × 2d(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∈Sym_d(ℝ) and N≥1, there exists such a frame satisfying lVert QX+PrVert_op ≤ c_d/N and lVert QrVert_op ≤ 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 ∑_q≥1q^ς−1ψ(q)^ς, ς= (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 ≥ 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.
ER -