Additive combinatorics

Erdős–Turán Conjecture on Arithmetic Progressions

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Must every infinite set of natural numbers with divergent reciprocal sum contain arithmetic progressions of every finite length? The source says yes is conjectured, not proved.

A \subseteq ℕ, \sum_{a \in A} a^{- 1} = \infty \Rightarrow A contains a k -AP for every k \geq 3.

Known results and sources

Divergent reciprocal mass is conjectured to force arithmetic progressions of every finite length; the distant target remains open.

Research problem

Exact mathematical statement

Let $A\subseteq\mathbb N$ be infinite. The conjecture is

\sum_{a\in A} \frac1a=\infty \quad\Longrightarrow\quad A\text{ contains a nonconstant }k\text{-term arithmetic progression for every }k\ge3.

Problem infographic

Problem at a glance

Positive positions carry accumulating reciprocal weights above three- and four-term equal-gap examples and an incomplete longer pattern. — The hypothesis, representative arithmetic progressions, and the unresolved all-length conclusion are shown without depicting a proof route.

Current mathematical picture

Where work on Erdős–Turán Conjecture on Arithmetic Progressions stands

Partially resolved

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureStationary finite-menu potential architectures

The source lists four rigorously dead stationary or decoupled architectures. Finite-horizon, structurally labelled, nonlinear, or scale-dependent lifting remains viable.

Route status · Narrowed route

Main reductionCurrent reduction

For each fixed length k, bound the harmonic sum H_k(T)=Σ_{m<T} α_k(3^m) uniformly in T.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Erdős–Turán Conjecture on Arithmetic Progressions in numbers

5.3kretained lines of mathematical investigation5,333 in the current working snapshot

Argument development: 4,473 · 84%
Explored or eliminated routes: 123 · 2%
Computational analysis: 105 · 2%
Open obligations: 326 · 6%
Definitions and setup: 306 · 6%

8selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPaul Erdős and Pál Turán (historical arithmetic-progression program; later reciprocal-sum form is conventionally associated with Erdős)Proposed1936Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.

Suggested move: Build a matrix- or vector-valued function on the periodic support from the exact fiber records and root embeddings.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

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Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeStationary finite-menu potential architectures

The source lists four rigorously dead stationary or decoupled architectures. Finite-horizon, structurally labelled, nonlinear, or scale-dependent lifting remains viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.Suggested move: Build a matrix- or vector-valued function on the periodic support from the exact fiber records and root embeddings.

Ready to work on

Close the short-parent S-B0 branch with a quantitative K_L bound or a label-based host argument.Suggested move: State the exact dependence on the logarithmic scale and insert it into the inherited terminal-scale budget.

Ready to work on

After a three-term closure, replace each cubic-only ingredient by an argument valid for every fixed length.Suggested move: Audit the Fourier identity, subgroup density cap, shell geometry, and list-transition bound one by one.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusPartially resolved

The reciprocal-sum conjecture is solved for three-term progressions through quantitative Roth-type bounds, but the assertion that every divergent-reciprocal-sum set contains progressions of every finite length remains open.

[4][2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Authoritative summaryGreen's JLMS centenary survey reviews the historical conjectures and modern quantitative landscape; it does not report an all-length reciprocal-sum solution.[4]
2023
Peer reviewedKelley and Meka proved a quasipolynomial-type upper bound for three-progression-free subsets, substantially strengthening the quantitative three-term theory.[3]
2020
PreprintBloom and Sisask crossed the logarithmic threshold needed to settle the first nontrivial reciprocal-sum case, k=3.[2]
1936
Historical sourceErdős and Turán introduced the extremal function r_k(N) and the density arithmetic-progression program that became foundational for the subject.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma2 of 82
equivalence1 of 81
computational claim1 of 81
negative result1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementDivergent reciprocal mass should force arithmetic progressions of all finite lengths.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementHarmonic reductionintermediate
retained route statementFinite transition checksintermediate
retained route statementFixed-three-term frontierintermediate
retained route statementAffine host branchintermediate
retained route statementStationary routes eliminatedintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThe finite transition values are retained only as a source-reported benchmark specialization. They do not establish, support, or prove the general reduction.specializes · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureStationary finite-menu potential architecturesreported failure
Research targetProve a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.open
Research targetClose the short-parent S-B0 branch with a quantitative K_L bound or a label-based host argument.open
Research targetAfter a three-term closure, replace each cubic-only ingredient by an argument valid for every fixed length.open
Research targetAll-length transfersuperseded
Narrowed routeStationary finite-menu potential architecturesThe source lists four rigorously dead stationary or decoupled architectures. Finite-horizon, structurally labelled, nonlinear, or scale-dependent lifting remains viable.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Prepared starting pointProve a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.

Erdős–Turán Conjecture on Arithmetic Progressions · ready to start

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Research contextPrepared context for any AI agent

Must every infinite set of natural numbers with divergent reciprocal sum contain arithmetic progressions of every finite length? The source says yes is conjectured, not proved.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
On Some Sequences of Integersoriginal source · Paul Erdős, Pál Turán · Journal of the London Mathematical Society 11, 261–264 · 1936 · DOI 10.1112/jlms/s1-11.4.261 · accessed Aug 14, 2026
2
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressionspreprint · Thomas F. Bloom, Olof Sisask · arXiv · 2020 · ARXIV 2007.03528 · accessed Aug 14, 2026
3
Strong Bounds for 3-Progressionspeer reviewed result · Zander Kelley, Raghu Meka · 64th IEEE Symposium on Foundations of Computer Science · 2023 · ARXIV 2302.05537 · DOI 10.1109/FOCS57990.2023.00059 · accessed Aug 14, 2026
4
Arithmetic progressions at the Journal of the LMSsurvey or monograph · Ben Green · Journal of the London Mathematical Society 113(3) · 2026-03-15 · DOI 10.1112/jlms.70483 · accessed Aug 14, 2026

Important qualifications

External metadata was collected independently of the unreviewed source material; packet claims are not external evidence.
Scoped searches do not establish that no additional result, formalization, or dataset exists elsewhere.
No external computation was rerun, and no source-submitted URL or packet attachment was fetched, executed, or rendered.
This metadata grants no proof, acceptance, credit, publication, redistribution, deployment, or public-write authority.

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Erdős–Turán Conjecture on Arithmetic Progressions

Exact mathematical statement

Problem at a glance

Where work on Erdős–Turán Conjecture on Arithmetic Progressions stands

Erdős–Turán Conjecture on Arithmetic Progressions in numbers

Prove a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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