Quantitative Oppenheim in signature $(2,2)$ via determinant values

Technical Summary: Quantitative Oppenheim in Signature (2, 2) via Determinant Values

Problem Statement
The paper addresses the quantitative Oppenheim problem for indefinite quadratic forms of signature $(2, 2)$. The classical Oppenheim conjecture, proved by Margulis, asserts that the values of an indefinite irrational quadratic form on integer vectors are dense in $\mathbb{R}$. The quantitative refinement asks for the asymptotic behavior of the counting function:
$$N_Q(a, b; T) := \#\{v \in \mathbb{Z}^n : \|v\| < T, a < Q(v) < b\}$$
as $T \to \infty$. While this was established for higher signatures by Dani–Margulis and Eskin–Margulis–Mozes (EMM), the low signatures $(2, 1)$ and $(2, 2)$ require additional Diophantine hypotheses to exclude lattices that are too well approximated by rational forms. The $(2, 2)$ case was previously proved by Eskin–Margulis–Mozes. This paper provides a new proof of the $(2, 2)$ case by recasting the problem in terms of determinant values on lattices in $M_2(\mathbb{R})$.

Methodology
The authors reformulate the problem using the identification of $\mathbb{R}^4$ with $M_2(\mathbb{R})$, where the quadratic form is the determinant:
$$Q_0 \begin{pmatrix} x & y \\ z & w \end{pmatrix} = xw - yz.$$
The problem is thus transformed into counting lattice points $\Lambda < M_2(\mathbb{R})$ with determinant values in a specific interval. The proof employs the modified-height and avoidance strategy, a dynamical approach previously developed in the authors' work on the $(2, 1)$ case.

Key methodological components include:

1. Dynamical Setup: The problem is analyzed on the space of unimodular lattices $X = SL_4(\mathbb{R})/SL_4(\mathbb{Z})$. The action of the group $H = SL_2(\mathbb{R}) \otimes SL_2(\mathbb{R})$ corresponds to the natural action on $M_2(\mathbb{R})$.
2. Modified Height Functions: The standard height function $\alpha(\Delta)$, which measures the size of the smallest non-zero vector in exterior powers of the lattice, fails to satisfy the required uniform moment bounds in the critical exterior degree 2 due to the presence of "quasi-null" rational planes. The authors introduce a modified height $\tilde{\alpha}_{\eta, M}$ that omits rational two-planes whose Plücker vectors are "quasi-null" (i.e., lie very close to the exceptional submodules $M_c$ or $M_r$).
3. Auxiliary Height and Contraction: To prove moment bounds for the modified height, an auxiliary height $\hat{\alpha}_{\eta, M, \theta}$ is introduced. This height incorporates a weight function $\phi_\theta$ that penalizes proximity to the exceptional submodules, compensating for the degeneration of contraction estimates near these subspaces.
4. Avoidance of Exceptional Sets: The proof utilizes a stopping-time iteration that combines:
   • Contraction estimates away from an exceptional set (where the auxiliary height contracts under the action of expanding horospherical subgroups).
   • Quantitative non-divergence and Diophantine conditions to show that the flow avoids the exceptional set (where a non-quasi-null rational plane is carried extremely close to an exceptional summand).
5. Separation of Singular Contributions: The authors distinguish between exact rational isotropic planes (where the determinant vanishes identically) and quasi-null planes. They prove that for determinant-irrational lattices, there are only finitely many exact rational isotropic planes. The contribution of quasi-null but non-isotropic planes is shown to vanish in the limit, while the exact isotropic planes contribute a singular term to the asymptotic formula.

Key Contributions and Results
The paper establishes the following main theorem (Theorem 1.1):

Let $\Lambda < M_2(\mathbb{R})$ be a Diophantine lattice that is not determinant-rational. For any norm $\|\cdot\|$ on $\mathbb{R}^4$ and any $a < b$, the number of lattice points with non-zero determinant satisfies:
$$N^\times_\Lambda(a, b; T) \sim \frac{C_{\|\cdot\|}}{\text{covol}(\Lambda)} (b-a)T^2.$$
Furthermore, the limit for points with zero determinant exists:
$$c^{\text{sing}}_\Lambda := \lim_{T \to \infty} T^{-2} \#\{v \in \Lambda : \|v\| < T, \det v = 0\}.$$
The total counting function is given by:
$$N_\Lambda(a, b; T) \sim \left( \frac{C_{\|\cdot\|}}{\text{covol}(\Lambda)}(b-a) + c^{\text{sing}}_\Lambda \mathbf{1}_{\{0 \in (a, b)\}} \right) T^2.$$

Specific Technical Findings:

• Finiteness of Isotropic Planes: For a non-determinant-rational lattice, there are at most four rational two-planes on which the determinant vanishes identically (at most two column-isotropic and two row-isotropic). This finiteness is a special feature of the $2 \times 2$ case.
• Singular Constant: The constant $c^{\text{sing}}_\Lambda$ is strictly positive if and only if $\Lambda$ contains a rank-two $\mathbb{Z}$-submodule on which the determinant vanishes. It is explicitly computed as a sum over these finite rational isotropic planes.
• Diophantine Condition: The result holds under a "split-Diophantine" condition, which prevents the determinant form from being approximated too closely by rational split quadratic forms. This is equivalent to the non-EWAS condition used by Eskin–Margulis–Mozes.
• Algebraic Coefficients: As a corollary, the theorem applies to any lattice with algebraic entries that is not determinant-rational, as these automatically satisfy the Diophantine condition via Liouville-type inequalities.

Significance and Claims
The paper claims its primary novelty lies in the proof technique rather than the statement of the theorem itself.

• Streamlined Model: By working in the $2 \times 2$ determinant setting, the authors provide a "streamlined model case" for the modified-height and avoidance method. In this dimension, the representation theory and sublevel estimates are elementary compared to higher-dimensional cases, yet the main dynamical mechanisms remain visible.
• New Proof of EMM: The work recovers the $(2, 2)$ quantitative Oppenheim theorem of Eskin–Margulis–Mozes. The authors note that their approach differs from the original EMM proof, which worked directly with $K$-averages and used shrinking/overlap arguments for bad sets. In contrast, this paper introduces a second auxiliary modified height to satisfy the required Margulis-type inequality, avoiding the direct approximation of bad sets by rectangles.
• Foundation for Higher Dimensions: The determinant formulation is presented as the simplest member of a higher-dimensional determinant-value problem studied in the authors' companion paper [7], suggesting that the insights gained here are foundational for tackling more complex signatures.

The paper does not propose new experimental applications or future implications beyond the mathematical context of dynamical systems and the quantitative theory of quadratic forms. It strictly focuses on establishing the asymptotic counting formula through the refined dynamical method.