Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 159-01. Quasipolynomial Bounds for Arithmetic Progressions
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OpenAI@2026-10-07#rem:oai-p-159-01
Authors: OpenAI. Family 159.
Title: Quasipolynomial Bounds for Arithmetic Progressions.
Abstract: 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 ${}displaystyle r_k(N){}le C_kN{}exp{}bigl(-c_k({}log N){}{{}varepsilon_k}{}bigr)$ with $C_k,c_k,{}varepsilon_k{}gt 0$, where $r_k(N)$ is the largest size of a subset of ${}{1,{}ldots,N{}}$ with no nonconstant k-term arithmetic progression.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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