Rectangles versus Isosceles Triangles in Lattice Sets
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions in the Mizohata-Takeuchi circle of problems.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Combinatorial geometry
- Posed by
- Jonathan Bennett and coauthors
- Year posed
- —
- Years open
- —
- Solved
- 2026-06-29
- Model
- ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Jonathan Bennett, Vjekoslav Kovac, Shohei Nakamura, Itamar Oliveira
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper says in its introduction, and again in a dedicated AI usage declaration, that the combinatorial construction of the finite lattice sets achieving the stated bounds was found by ChatGPT 5.5 Pro; the authors ran it repeatedly and developed the surrounding estimates. Gemini 3.1 Pro produced the TikZ for the figures.
Verification
arXiv preprint; not yet peer-reviewed.
Source
arXiv:2606.30178 - Rectangles, triangles and Schrodinger waves