Problem detail · source-aware
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Near-optimal density thresholds forcing a measurable set in $\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
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'.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
No verification note supplied.
Correctness: unknown · statement fidelity: unaudited · peer review: none
Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.