On the paucity of lattice triangles

Abstract: A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface $(X_T,ω_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,ω_T)$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of $(X_T,ω_T)$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(π/2,2π/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a density 0 subset of the triangles in this window. The main engine in this paper was autoformalized by AxiomProver in Lean (using mathlib).

• The paper establishes that lattice triangles in the hard obtuse regime are sparse, with their density tending to zero for most candidate denominators.
• It employs advanced Fourier analytic decomposition and Ramanujan sums to rigorously bound error terms and verify the Mirzakhani–Wright rank obstruction.
• The methodology is formally verified using AxiomProver, highlighting the integration of number theory, harmonic analysis, and automated proof checking.

Quantitative Density Obstructions for Lattice Triangles in the Hard Obtuse Window

Introduction

The study of lattice triangles—rational triangles whose translation surfaces exhibit the Veech or “lattice” property—occupies a central position in translation surface dynamics and Teichmüller theory. These surfaces arise in the unfolding of Euclidean billiards in rational polygons and have closed $\mathrm{SL}_2(\mathbb{R})$-orbits in moduli space, giving rise to Teichmüller curves. While acute and right-angled rational triangles are fully classified, the obtuse scalene regime, particularly for triangles with the largest angle in $(\pi/2, 2\pi/3]$, remains a primary unresolved domain.

This paper provides a fine-grained quantitative analysis of the "paucity" of lattice triangles in this hard obtuse regime and leverages recent advances in number-theoretic reformulations of the Mirzakhani--Wright rank obstruction. The main theorem shows that, for a density 1 subset of denominators, the proportion of lattice triangles tends to zero, yielding overwhelmingly negative answers to the lattice property among candidates in the hard window.

Lattice Triangle Problem and Its Reduction

A rational triangle $T$ with interior angles $(p\pi/n, q\pi/n, r\pi/n)$ (in lowest terms, with $p+q+r=n$ and $\gcd(p,q,r,n)=1$) unfolds, via finitely many reflections, into a compact translation surface $(X_T, \omega_T)$. The key question is to classify those $T$ whose unfolding is a Veech (lattice) surface, with an affine automorphism group that is a lattice in $\mathrm{SL}_2(\mathbb{R})$, inducing a Teichmüller curve in moduli space.

In acute and right-angled cases, the classification is essentially complete; the only infinite families of obtuse lattice triangles are those having angles $(\frac{\pi}{n}, \frac{\pi}{n}, \frac{(n-2)\pi}{n})$, $$(\frac{\pi}{2n}, \frac{\pi}{n}, \frac{(2n-3)\pi}{2n})$$, and a sporadic example due to Hooper. The prevailing conjecture, supported by experimental and theoretical evidence, is that no further obtuse rational lattice triangles exist.

The “hard window” is identified as the region where the largest angle lies in $(\pi/2, 2\pi/3]$. Prior work has shown that for angles $>2\pi/3$, the number-theoretic form of the Mirzakhani--Wright obstruction rules out nearly all triangles, but the hard window remained elusive.

Number-Theoretic Criteria and Main Results

Larsen--Norton--Zykoski provided a crucial reduction: the Mirzakhani--Wright geometric obstruction is equivalent to verifying the existence of a “usable” unit $a\in (\mathbb Z/n\mathbb Z)^\times$ for which certain modular conditions involving the angles are satisfied. Specifically, if at least two of $[ap]_n < [2p]_n$, $[aq]_n < [2q]_n$, $[ar]_n < [2r]_n$ hold, the rank obstruction blocks the lattice property.

For a fixed $n$, consider $$\mathcal{H}_n = \{ (p, q) \in \mathbb{Z}_{\ge 1}^2 : p+q < n/2, \gcd(p,q,n) = 1 \}$$—the set of candidate angle pairs in the obtuse regime. The central result is as follows: for $n$ with its largest prime divisor $P^+(n)\ge n^{1/\log\log n}$ (which occurs for a density 1 subset by Dickman–de Bruijn theory), the density of lattice triangles in $\mathcal{H}_n$ tends to zero as $n\to\infty$.

The main technical engine is the Fourier-analytic decomposition of a counting function $S(p,q)$, which encodes the number of units $a$ satisfying the modular inequalities. By analyzing the error terms with restricted Ramanujan sums and leveraging the presence of large prime divisors, the authors show that for almost all $(p,q)$, the cardinality $S(p,q)$ is sufficiently large to guarantee the rank obstruction, thereby precluding the lattice property.

Analytical Structure and Methodology

The core analytic construction is the decomposition

$S(p,q) = M(p,q) + E(p,q)$

where $M(p,q)$ is the main term capturing average behavior, and $E(p,q)$ is an error term involving sums of Ramanujan-type exponential sums over units. Detailed Fourier analysis of indicator functions for short intervals and careful bounding of the corresponding Fourier coefficients enable explicit error estimates.

The presence of a large prime factor in $n$ is pivotal. For $n = P^\alpha m$ (with $P$ prime and $\gcd(m,P)=1$), the Ramanujan sum $c_{P^\alpha}(t)$ vanishes unless $P^{\alpha-1} \mid t$, and for many $(p,q)$, the error terms over exceptional residue classes are sufficiently sparse to be negligible. These estimates are then consolidated with density arguments and divisor bounds to show the overwhelming failure of the lattice condition in the hard window.

Formal Verification and Computational Aspects

A notable aspect of the work is the autonomous formalization of the main analytic engine in Lean (via AxiomProver). This verifies, at the formal proof-assistant level, the combinatorial and analytic arguments needed for the density bounds, showcasing the maturity of proof automation in modern mathematical research. The theorem formalized states that, for any $\eta\in (0,1/6)$ and $\theta\in (0,1)$, as $n\to\infty$ with $P^+(n) \ge n^\theta$, the negative obstruction property holds for all but a vanishing proportion of candidate pairs in $\mathcal{H}_n(\eta)$.

Implications, Limitations, and Future Directions

The demonstrated density-1 obstruction for the lattice property in the hard obtuse window provides near-complete evidence for the conjectural classification of lattice triangles. Practically, it implies that, outside a small exceptional set (predominantly $n$ with only small prime factors), the search and identification of new obtuse lattice triangles in the hard window is fruitless, and that sporadic or exceptional examples are vanishingly rare in the natural parameterization.

Theoretically, this work highlights the effectiveness of number-theoretic and harmonic-analysis-based techniques for translation surface classification problems, as well as the power of Mirzakhani–Wright-type rank obstructive arguments. The explicit error control achieved in this context is much sharper than what is available for most geometric or arithmetic dynamical classification questions.

On the computational and formal end, the ability to use automated systems (AxiomProver) for the generation and verification of core analytic machinery suggests promising directions for AI-assisted proof checking and even discovery in arithmetic dynamics and Teichmüller theory.

Open directions include the full classification of the vanishingly small exceptional set (potentially involving deeper arithmetic geometry or Diophantine methods), delineation of further structures in the moduli space of translation surfaces, and possible extensions of the quantitative techniques to related billiard or polygonal unfolding problems in higher-dimensions or with more general symmetry types.

Conclusion

This analysis establishes that the Mirzakhani–Wright rank obstruction, when recast in explicit number-theoretic terms and analyzed with detailed Fourier techniques, suffices to show that lattice triangles in the hard obtuse regime constitute a set of density zero among all rational triangles. This not only provides strong evidence for the conjectural completeness of the known list of obtuse lattice triangles but also sets a methodological blueprint for similar problems at the intersection of flat geometry, arithmetic dynamics, and formal verification in mathematics.