Remark: Family 159. Erd (o)s's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

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 $k\ge 3$, every subset of $\{1,\dots ,N\}$ with no nonconstant $k$-term progression has size at most $C_{k}N\exp [-c_k(\log N{)}^{{\epsilon }_k}]$, with positive constants depending only on $k$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.