Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

What OpenAI claims

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 ≥ 3, every subset of {1,…,N} with no nonconstant k-term progression has size at most C_kN exp[-c_k(log N)^ε_k], with positive constants depending only on k. (Lean)

Paper

1. Quasipolynomial Bounds for Arithmetic Progressions

   23 September 2026 · Preprint, not peer reviewed · PDF

   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

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.