OpenAI · 084-02 · 2026-09-25 · Real and complex analysis
The dyadic case of the Erdős similarity conjecture
preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026
Family 084. The geometric case of the Erdős similarity 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
For every fixed , constructs compact subsets of with measure arbitrarily close to one containing no translated and nontrivially dilated copy of , with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.
We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence . The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 084: The geometric case of the Erdős similarity 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.