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Distribution of values at tuples of integer vectors under symplectic forms

Jiyoung Han

TL;DR

This work extends the Oppenheim conjecture to high-rank symplectic forms by leveraging homogeneous dynamics, Ratner’s theorems, and the maximality of ${\rm Sp}(2n,\mathbb{R})$ in ${\rm SL}_{2n}(\mathbb{R})$ to obtain qualitative density criteria for $k$-tuples of integer vectors. It then derives a sharp volume formula for the relevant rank-$k$ cone, establishing the expected growth rate $T^{2nk-k(k-1)}$ and enabling asymptotic counting via Rogers' higher-rank Siegel transform formulas. The authors prove a random quantitative Oppenheim-type result for generic symplectic forms and extend the framework to primitive and congruence variants, showing that quantitative statements hold even without full explicit higher-moment formulas in the primitive setting. Overall, the paper advances the quantitative understanding of lattice-point distributions under symplectic forms, providing tools for equidistribution and lattice counting in high-rank homogeneous dynamics settings.

Abstract

We investigate lattice-counting problems associated with symplectic forms from the perspective of homogeneous dynamics. In the qualitative direction, we establish an analog of Margulis theorem for symplectic forms, proving density results for tuples of vectors. Quantitatively, we derive a volume formula having a certain growth rate, and use this and Rogers' formulas for a higher rank Siegel transform to obtain the asymptotic formulas of the counting function associated with a generic symplectic form. We further establish primitive and congruent analogs of the generic quantitative result. For the primitive case, we show that the lack of completely explicit higher moment formulas for a primitive higher rank Siegel transform does not obstruct obtaining quantitative statements.

Distribution of values at tuples of integer vectors under symplectic forms

TL;DR

This work extends the Oppenheim conjecture to high-rank symplectic forms by leveraging homogeneous dynamics, Ratner’s theorems, and the maximality of in to obtain qualitative density criteria for -tuples of integer vectors. It then derives a sharp volume formula for the relevant rank- cone, establishing the expected growth rate and enabling asymptotic counting via Rogers' higher-rank Siegel transform formulas. The authors prove a random quantitative Oppenheim-type result for generic symplectic forms and extend the framework to primitive and congruence variants, showing that quantitative statements hold even without full explicit higher-moment formulas in the primitive setting. Overall, the paper advances the quantitative understanding of lattice-point distributions under symplectic forms, providing tools for equidistribution and lattice counting in high-rank homogeneous dynamics settings.

Abstract

We investigate lattice-counting problems associated with symplectic forms from the perspective of homogeneous dynamics. In the qualitative direction, we establish an analog of Margulis theorem for symplectic forms, proving density results for tuples of vectors. Quantitatively, we derive a volume formula having a certain growth rate, and use this and Rogers' formulas for a higher rank Siegel transform to obtain the asymptotic formulas of the counting function associated with a generic symplectic form. We further establish primitive and congruent analogs of the generic quantitative result. For the primitive case, we show that the lack of completely explicit higher moment formulas for a primitive higher rank Siegel transform does not obstruct obtaining quantitative statements.

Paper Structure

This paper contains 7 sections, 26 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Oppenheim Conjecture for Symplectic Forms
  3. Volume Formula
  4. Use of Rogers' Formulas and Proof of Theorem \ref{['thm: quantitative']}
  5. Primitive and Congruent Analogs
  6. Proof of Theorem \ref{['thm: quantitative prime']}
  7. Proof of Theorem \ref{['thm: quantitative congruence']}

Key Result

Theorem 1.1

Let $n\ge 2$. Assume that $2\le k\le n+2$ for $n\ge 3$ and $k=2,\;3$ when $n=2$. Let $\langle \cdot, \cdot \rangle^g$ be a symplectic form on ${\mathbb{R}}^{2n}$ associated with $g\in \mathrm{SL}_{2n}({\mathbb{R}})$. The image set is dense in ${\mathbb{R}}^{\frac{1}{2} k(k-1)}$ if and only if $\langle \cdot, \cdot \rangle$ is not a scalar multiple of any symplectic forms with integer coefficients

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2: Ratner's Orbit Closure Theorem
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • ...and 30 more