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Rectangles versus Isosceles Triangles in Lattice Sets

Combinatorics · posed by Jonathan Bennett and coauthors · disproved

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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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Jonathan Bennett, Vjekoslav Kovac, Shohei Nakamura and Itamar Oliveira, using ChatGPT 5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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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Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.