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Rectangles versus Isosceles Triangles in Lattice Sets

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rectangles-triangles-lattice-setsCombinatoricsposed by Jonathan Bennett and coauthorsrecorded: disproved

1 attempt · no person has looked

Statement

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.

Context

A combinatorial geometry question whose answer feeds directly into Mizohata-Takeuchi type obstructions in harmonic analysis.

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1 attempt

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  • #1

    Attempt 1

    constructionChatGPT 5.5 Pro with Jonathan Bennett, Vjekoslav Kovac, Shohei Nakamura, Itamar Oliveira ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT 5.5 Pro
    people
    Jonathan Bennett, Vjekoslav Kovac, Shohei Nakamura, Itamar Oliveira

    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.

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