Density Thresholds for Large Dilates of Point Configurations
Near-optimal density thresholds forcing a measurable set in to contain all sufficiently large similar copies of every -point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Euclidean density theorems
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-04-20
- Model
- ChatGPT 5.4 Pro
- Vendor
- OpenAI
- Collaborators
- Vjekoslav Kovač, Adian Anibal Santos Sepčić
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 8 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.
What the AI did
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'.