Heilbronn's triangle problem

OpenAI's theorem (2026) that n points in the unit square can have every triangle of area at least a constant times n^(-2+eta) for a fixed eta > 0; through field norms written in a large base, residue orbits, a cap at an auxiliary prime, lattice counts and a deletion argument.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Small triangles and the plan
2. Counting lattice matrices
3. A determinant obstruction from field norms
4. Residue matrices and their orbits
5. A cap at an auxiliary prime
6. Sampling integer columns
7. Determinant zero
8. Deletion and the theorem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Thom classes and Euler classes · Characteristic classes
• Finite fields and group actions (core courses Abstract Algebra I and II), Lebesgue measure (core course Measure and Integration), compactness (core course Real Analysis II), and Lemma 5.1 of Thom classes and Euler classes on subgroups of free abelian groups.