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Density Thresholds for Large Dilates of Point Configurations

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Statement

Near-optimal density thresholds forcing a measurable set in Rd\mathbb{R}^d to contain all sufficiently large similar copies of every nn-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.

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

  1. proof attempt · #1

    Vjekoslav Kovač and Adian Anibal Santos Sepčić, using ChatGPT 5.4 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 assisted
    — a person led the work and used a model along the way.

    ChatGPT 5.4 Pro "was used to suggest and draft approaches to Proposition 5"; in particular the random pattern thinning argument, "somewhat novel in this context," was suggested by the model. Main ideas and final proofs are the authors'.

    Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.

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