OpenAI · 073-01 · 2026-09-23 · Real and complex analysis
The Falconer distance conjecture in all dimensions
preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026
Family 073. The Falconer distance conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Resolves the Falconer distance conjecture in every dimension : every compact set with Hausdorff dimension greater than determines a set of Euclidean distances of positive Lebesgue measure.
We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than determines a set of Euclidean distances of positive Lebesgue measure.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 073: The Falconer distance conjecture. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.