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On the paucity of lattice triangles

David Kurniadi Angdinata, Evan Chen, Ken Ono, Jiaxin Zhang, Jujian Zhang

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).

On the paucity of lattice triangles

Abstract

A rational triangle (one whose angles are rational multiples of ) unfolds to a translation surface . The lattice triangle problem asks to classify those for which is a Veech (lattice) surface, which means that the -orbit of 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 ), 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).
Paper Structure (24 sections, 11 theorems, 109 equations)

This paper contains 24 sections, 11 theorems, 109 equations.

Table of Contents

  1. Introduction and statement of results
  2. Number theoretic form of the Mirzakhani--Wright rank obstruction
  3. The sum $S(p,q)$ and its Fourier expansion into Ramanujan sums
  4. Discrete Fourier coefficients of short intervals
  5. Ramanujan sums
  6. The main term and error term for S(p,q)
  7. Utility of large prime factors
  8. Splitting the error term by P-adic valuation
  9. Bounding the restricted sum
  10. Proof of Proposition \ref{['prop:errorBound']}
  11. A crude bound for $E_{P_0}$.
  12. Reduction of $E_{P^\alpha}$ to a residue-class sum.
  13. Averaging over the unit $u\equiv -p q^{-1}\pmod d$.
  14. Conclusion.
  15. Proof of Theorem \ref{['thm:MainQuantitative']} and Corollary \ref{['cor:MainDensity']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

For $n \in \Omega^+$, let $\mathcal{L}_n \subset \mathcal{H}_n$ denote the set of pairs $(p,q)$ corresponding to lattice triangles $(p\pi/n, q\pi/n, (n-p-q)\pi/n)$. Then we have that

Theorems & Definitions (22)

  • Example
  • Remark
  • Conjecture
  • Theorem 1.1
  • Corollary 1.2
  • Remark
  • Proposition 2.1: Larsen--Norton--Zykoski, Proposition 2.1 LNZ
  • Lemma 3.1: Fourier coefficients of an interval
  • proof
  • Lemma 3.2: Ramanujan sums at prime powers
  • ...and 12 more