Singular systems of linear forms over global function fields

1. Problem Statement

The paper addresses the problem of estimating the Hausdorff dimension of the set of singular systems of linear forms over global function fields.

• Context: In Diophantine approximation, a matrix $A \in M_{m,n}(\mathbb{R})$ is called singular if the system of linear forms $A\mathbf{q} + \mathbf{p}$ can be approximated arbitrarily well by integers for large $T$. Specifically, for any $\epsilon > 0$, there exists $T_0$ such that for all $T \geq T_0$, there exist $\mathbf{p} \in \mathbb{Z}^m$ and $\mathbf{q} \in \mathbb{Z}^n$ satisfying $\|A\mathbf{q} + \mathbf{p}\| \leq \epsilon T^{-n/m}$ and $0 < \|\mathbf{q}\| \leq T$.
• Previous Work: Over real numbers ($\mathbb{R}$), the Lebesgue measure of singular matrices is zero (Khintchine, 1926). The computation of their Hausdorff dimension was a major open problem until recently. Das, Fishman, Simmons, and Urbański (2024) proved the sharp upper bound for real matrices: $\dim_H \text{Sing}(m,n) = mn - \frac{mn}{m+n}$.
• Gap: While results exist for specific cases over function fields (e.g., singular pairs), a general bound for $m \times n$ singular matrices over global function fields was missing.
• Goal: To prove that the Hausdorff dimension of the set of singular matrices over global function fields is bounded above by $mn - \frac{mn}{m+n}$, matching the real case.

2. Mathematical Setting

The authors work within the framework of global function fields:

• Let $K$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$.
• Fix a discrete valuation $\nu$ on $K$. Let $K_\nu$ be the completion (isomorphic to Laurent series $\mathbb{F}_{q_\nu}((\pi_\nu))$) and $O_\nu$ its valuation ring.
• Let $R_\nu$ be the affine algebra of the curve $C \setminus \{\nu\}$.
• Assumption: The authors assume $R_\nu$ is a Principal Ideal Domain (PID). This is crucial for the submodularity of the covolume function used in their proofs.
• Definition of Singularity: A matrix $A \in M_{m,n}(K_\nu)$ is singular if for any $\epsilon > 0$, there exist $\mathbf{p} \in R_\nu^m$ and $\mathbf{q} \in R_\nu^n$ satisfying the singular approximation condition.

3. Methodology

The paper employs the Dani Correspondence, linking Diophantine approximation to homogeneous dynamics.

A. Homogeneous Dynamics Setup

• Group: $G = SL_{m+n}(K_\nu)$ and lattice $\Gamma = SL_{m+n}(R_\nu)$.
• Space: $X = G/\Gamma$, identified with the space of unimodular lattices in $K_\nu^{m+n}$.
• Flow: A diagonal flow $g_t = \text{diag}(\pi_\nu^{-nt} I_m, \pi_\nu^{mt} I_n)$ acts on $X$.
• Correspondence: A matrix $s$ is singular if and only if the trajectory $\{g_t u_s \Gamma\}_{t \geq 1}$ diverges in $X$ (leaves every compact set), where $u_s$ is the unipotent subgroup associated with $s$.

B. Margulis Functions and Contraction

To estimate the dimension of the divergent set, the authors construct a Margulis function (a height function) that contracts on average along the flow.

1. Linear Algebra Estimates (Sections 3 & 4):
   • They establish contraction properties for integrals of the form $\int \|g_t u_s v\|^{-\beta} d\mu(s)$.
   • Key Innovation: Unlike the real case (Das et al.) which uses Gaussian measures, the authors use the restricted Haar measure on $O_\nu^{mn}$. They prove that analogous contraction properties hold for the Haar measure in the non-Archimedean setting.
   • They utilize Exterior Algebra and the Hodge Dual Operator ($\ast$). A significant technical hurdle is that the Hodge dual is not equivariant under the orthogonal group $SL_d(O_\nu)$ in the non-Archimedean setting. They overcome this by applying duality to horospherical subgroups and proving specific identities (Lemma 4.2) to handle the lack of equivariance.
2. Submodularity (Section 5):
   • They define a height function $\alpha_i(x)$ based on the norm of the wedge product of basis vectors of rational subspaces.
   • Using the assumption that $R_\nu$ is a PID, they utilize the submodularity of the covolume function (established in a previous work by Bang, 2025) to derive a contraction inequality for the Margulis function $\mathcal{R}(x) = \sum (\omega_i \alpha_i(x))^{\beta_i}$.
   • Result: They prove that for large $t$, the average value of $\mathcal{R}(g_t u_s x)$ is strictly less than $\mathcal{R}(x)$, implying that trajectories staying in the "singular" region (where $\mathcal{R}$ is large) are rare.

C. Covering Estimates (Section 6)

• Using the contraction of the Margulis function, they derive bounds on the measure of sets where the trajectory remains "large" (escaping on average).
• They employ a convolution argument (inspired by Guan and Shi, 2020) adapted to the ultrametric setting. They show that the convolution of Haar measures on $O_\nu$ behaves similarly to the real case, allowing them to bound the number of balls needed to cover the set of singular matrices.
• They prove that the set of points that "escape on average" (a generalization of singular trajectories) has a Hausdorff dimension bounded by $mn - \delta \frac{mn}{m+n}$.

4. Key Contributions

1. Extension to Function Fields: The paper successfully extends the breakthrough result of Das, Fishman, Simmons, and Urbański (2024) from real numbers to global function fields.
2. Haar Measure vs. Gaussian: The authors replace the Gaussian measure techniques used in the real case with restricted Haar measures on $O_\nu$, proving that the necessary contraction estimates hold in the non-Archimedean setting.
3. Handling Hodge Duality: They resolve the technical issue of the non-equivariance of the Hodge dual operator under $SL_d(O_\nu)$ by utilizing specific properties of horospherical subgroups and exterior algebra over function fields.
4. Submodularity Assumption: The proof relies on the assumption that the ring of integers $R_\nu$ is a PID. This allows the use of submodularity of the covolume, a key step in constructing the Margulis function.

5. Main Results

Theorem 1.1:
Assume $R_\nu$ is a principal ideal domain. For any $m, n \in \mathbb{Z}_{\geq 1}$, the Hausdorff dimension of the set of singular matrices $\text{Sing}(m,n)$ over $K_\nu$ satisfies:
$$\dim_H \text{Sing}(m,n) \leq mn - \frac{mn}{m+n}$$
This matches the known sharp upper bound for the real case.

Proposition 6.5 (Stronger Result):
The authors prove a stronger statement regarding the dimension of the set of lattices whose trajectories "escape on average" with rate $\delta$:
$$\dim_H \{ s \in M_{m,n}(K_\nu) : u_s x_0 \text{ } \delta\text{-escapes on average} \} \leq mn - \delta \frac{mn}{m+n}$$
Since singular matrices correspond to the case $\delta = 1$, this implies Theorem 1.1.

6. Significance

• Completeness of Theory: This work fills a critical gap in the theory of Diophantine approximation over function fields, showing that the geometric and dynamical behavior of singular systems is analogous to the real case, despite the different analytic tools required (ultrametric vs. Archimedean).
• Methodological Bridge: It demonstrates how techniques from homogeneous dynamics (Margulis functions, Dani correspondence) can be adapted to non-Archimedean settings, specifically addressing the challenges of measure theory and linear algebra in function fields.
• Future Directions: The result suggests that the "sharpness" of the bound (i.e., whether the dimension is exactly $mn - \frac{mn}{m+n}$) likely holds for function fields as well, pending the construction of a matching lower bound, which is a natural next step for the field.

In summary, the paper provides a rigorous proof that the Hausdorff dimension of singular systems of linear forms over global function fields (with class number one) is bounded by the same formula as in the real case, achieved through a sophisticated adaptation of dynamical methods to the non-Archimedean context.