Result 159, Combinatorics

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,…,N}\{1,\ldots,N\} with no nonconstant k-term progression has size at most CkNexp⁡[−ck(log⁡N)εk]C_kN\exp[-c_k(\log N)^{\varepsilon_k}], with positive constants depending only on k.

Lean formalization Reasoning summary (PDF) Proof

The bigger picture

Why it matters

Can a sparse collection of positive integers still be forced to contain arbitrarily long, evenly spaced patterns? The manuscript claims that it can whenever the sum of those integers' reciprocals grows without bound.

What changes?

An arithmetic progression is a sequence with a constant gap; nonconstant means that gap is nonzero. For every fixed length k at least 3, the unreviewed manuscript reports that a subset of 1 through N containing no such progression has size at most C times N times exp(-c (log N)^epsilon). The constants C, c and epsilon are positive and depend only on k. This gives an explicit size restriction, with no asserted uniformity across progression lengths.

What does that help mathematicians do?

The bound would let researchers rule out any infinite set that simultaneously avoids a fixed progression length and has a divergent reciprocal sum. Grouping integers into successively doubled ranges explains the connection: the size restriction makes those ranges' reciprocal contributions decay fast enough to have a finite total. Thus divergence forces every finite progression length, even without assuming that the set occupies a positive fraction of the integers.

Are there practical applications?

Its immediate value is foundational: it links a numerical measure of an integer set's abundance to unavoidable additive patterns. The quantitative bound also limits how large finite progression-free examples can be. The abstract supplies neither numerical values for the length-dependent constants nor a practical procedure for finding the progressions, so concrete computational benefits remain unestablished.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Quasipolynomial Bounds for Arithmetic Progressions

September 23, 2026 198 pages

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

rk(N)≤CkNexp⁡(−ck(log⁡N)εk)\displaystyle r_k(N)\le C_kN\exp\bigl(-c_k(\log N)^{\varepsilon_k}\bigr)

with Ck,ck,εk>0C_k,c_k,\varepsilon_k\gt 0, where rk(N)r_k(N) is the largest size of a subset of {1,…,N}\{1,\ldots,N\} with no nonconstant k-term arithmetic progression.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026,
  author = {{OpenAI}},
  title = {{Quasipolynomial Bounds for Arithmetic Progressions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf}{OAI:Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/159.md.

Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Erdős's reciprocal-sum conjecture asks whether every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. The formalization proves this statement: for every requested length, such a set contains a progression with positive common difference.

The selected theorem is the reciprocal-sum consequence. The paper's quantitative upper bound for the largest progression-free subset of {1,…,N}\{1,\ldots,N\} is outside this statement.

Comparator links

Result Comparator statement
Erdős's reciprocal-sum arithmetic-progression conjecture ErdosReciprocal.lean

Posts about this result

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.