OpenAI · 159-01 · 2026-09-23 · Combinatorics
Quasipolynomial Bounds for Arithmetic Progressions
preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026
Family 159. Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds.
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
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed , every subset of with no nonconstant -term progression has size at most , with positive constants depending only on .
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show with , where is the largest size of a subset of with no nonconstant k-term arithmetic progression.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 159: Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds. 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.