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Erdős’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 ≥ 3, every subset of

\{1,\ldots,N\}

with no nonconstant k-term progression has size at most

C_kN\exp[-c_k(\log N)^{\varepsilon_k}]

, with positive constants depending only on k.

Manuscripts in this family · 1

Quasipolynomial Bounds for Arithmetic Progressions

Sep 23, 2026·paper.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

\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.

Full abstract

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OpenAI also released the model's reasoning trace for this result: reasoning trace (PDF).