Number theory · additive bases · representation functions

Erdős–Turán Additive-Basis Conjecture

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If a set of nonnegative integers represents every sufficiently large integer as a sum of two of its elements, must some integers have arbitrarily many ordered representations?

r_{A} (n) > 0 eventually \Rightarrow \sup_{n} r_{A} (n) = \infty

Erdős Problem #28

Known results and sources

The conjecture asks whether eventual pair-sum coverage can coexist with uniformly bounded multiplicity; the current packet-split route still needs a valid moving-tail or terminal conversion.

Research problem

Exact mathematical statement

For a set $A\subseteq\mathbb N_0$ , let the ordered representation function be

r_A(n)=\#\{(a,b)\in A^2:a+b=n\}.

The conjecture says that

r_A(n)>0\text{ for all sufficiently large }n \quad\Longrightarrow\quad \sup_n r_A(n)=\infty.

In words, an asymptotic basis of order two cannot have a uniformly bounded ordered representation function. The current source reports a strong one-scale packet-split reduction, but it has not promoted that chain uniformly to moving tails or converted its typed correlation into a contradiction. The conjecture remains open, and ProofAtlas has not reproduced the submitted computations or upgraded the source's workstream labels to independently checked evidence.

Problem infographic

Problem at a glance

A selected set of integer points feeds an ordered-pair grid whose constant-sum diagonals contain different numbers of glowing representations, beside a long covered sequence with an unresolved multiplicity horizon. — Every sufficiently large output must be represented by a pair from the set; the open question is whether the number of ordered pairs must become arbitrarily large.

Current mathematical picture

Where work on Erdős–Turán Additive-Basis Conjecture stands

Open conjecture

The cumulative v12 source keeps the Erdős–Turán conjecture open. Its current primary route is ET-R12 packet-split renewal; ET-R7 remains an active inherited interface within the retained ET-C0 content split, not the primary route.

Leading routeET-R12 packet-split renewal through moving tails — current primary route

Current primary route: promote the packet-split chain to moving tails, then compare lifetime-disjoint scales for recurrent kappa, a common descendant, an independent cube direction, or typed first-use excess.

Route status · Active route

Main reductionCoherent finite-to-Abel bridge remains a valid fallback

The source retains the route from translated-prefix recurrence through coherent cap-five near-bases to the Abel hole contradiction, while preferring the stronger finite cap-three cube endpoint whenever separated low-loss switches can be extracted.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeMoving-tail promotion of the v12 packet-split chain

Promote ET-AFF7--ET-PKT7 uniformly from the normalized exact basis to moving tails, preserving ordinary-hole deletion, positivity, output ranges, typed thresholds, and exact lifetimes.

Task status · Ready to work on

Research-record correctionResearch-record correction

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 Additive-Basis Conjecture in numbers

4.1kretained lines of mathematical investigation1,225 in the current working snapshot

Argument development: 3,447 · 83%
Explored or eliminated routes: 82 · 2%
Computational analysis: 96 · 2%
Open obligations: 303 · 7%
Definitions and setup: 206 · 5%

17inventoried working statements15routes investigated8open questions8contribution-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, Pál TuránProposed1941Status checkedAug 26, 2026

Source recordedAug 30, 2026

Recommended next task

Moving-tail promotion of the v12 packet-split chain

Promote ET-AFF7--ET-PKT7 uniformly from the normalized exact basis to moving tails, preserving ordinary-hole deletion, positivity, output ranges, typed thresholds, and exact lifetimes.

Suggested move: Prove the moving-tail analogue of ET-AFF7 and audit forward and backward packet holes through complete ET-MT2 lifetime intervals.

What would count as progress

Uniform constants for all M<=theta X and explicit deletion of ordinary tail holes before coverage.
Every packet, rank, color, diagonal deletion, output range, and lifetime remains in the typed signature.

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

28 selected steps

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

Scroll horizontally to explore the route

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.

Active routeET-R12 packet-split renewal through moving tails — current primary route

Route status · Active route

Active routeET-R7 inherited outermost-star/product-core renewal

Retained active inherited interface: renew outermost-star and product-core structure across separated scales. In v12 this remains supporting work within the inherited ET-C0 content split; it is not the current primary route.

Route status · Active route

Active routeET-R8-MODULAR-SYNC prime descendants meet stars and transports

15.2 Route ET-R8-MODULAR-SYNC — prime descendants meet stars and transports Use ET-A9 in the same moving tails as ET-MT3 and ET-GRID1. Prime separation prevents one localized gap from serving too many large moduli. Seek a double-counting theorem over (modulus, star edge, shift, grid cell, transport) that forces a recurrent gap or many independent increments. This route may eliminate the pinned matching by imposing arithmetic structure on its gap set.

Route status · Active route

Active routeET-R9-FIRSTUSE exact typed-witness first use

15.3 Route ET-R9-FIRSTUSE — exact typed-witness first use Define first use only after quotienting the exact ET-MT2 lifetime interval of a fully typed physical witness. ET-CAN3 gives the exact excess term for repeated witnesses. Prove bounded separated-scale reuse unless a recurrent lower-cap object appears, or attach every new use to a new independent increment. Shared prefixes and arbitrary fragment restrictions require no additional theorem.

Route status · Active route

Active routeET-R4-NEARBASIS lower-cap finite recurrence

15.4 Route ET-R4-NEARBASIS — lower-cap finite recurrence Use the cap-halving core to produce a finite lower-cap near-basis and then a coherent Abel limit. ET-T18C and ET-TERM1 are the endpoints. The missing quantitative requirement is a hole error small enough relative to sqrt W and a diameter-matched target interval.

Route status · Active route

Active routeET-R3-CAPTURE terminal interval compression

15.5 Route ET-R3-CAPTURE — terminal interval compression Develop a density increment or interval-selection theorem for C+C, informed by the pinned-star exclusion sets. Repeatedly restrict to a subinterval where represented-target density increases while preserving coordinate mass and cap. Terminate at ET-TERM1 or pay ET-T25’s H,O,C,T,E currency.

Route status · Active route

Active routeET-R1-SCI / ET-R6-SPECTRAL-CAPTURE quantitative diagonal route

15.6 Route ET-R1-SCI / ET-R6-SPECTRAL-CAPTURE — quantitative diagonal route Retain ET-T29 to relate a bad short-difference window to a mesoscopic packet. ET-S2 closes interior certification. The remaining analytic inverse theorem should correlate one common shift or modular descendant with the packet, or directly yield ET-T4D. This is the strongest planned route change if structural renewal stalls.

Route status · Active route

Active routeET-R2-IDEM direct zero-one square deficit

15.7 Route ET-R2-IDEM — direct zero-one square deficit Exploit simultaneously that F has zero-one coefficients, F^2 has bounded nonnegative coefficients, support is one-sided, and the Fejér functional is large. Generic norm and sign arguments are exhausted; a successful proof needs a new idempotent or uncertainty principle adapted to bounded convolution coefficients. ---

Route status · Active route

Explored alternatives

Other routes

7 recorded

Useful but insufficientOne-window or repeatedly recycled product-scale charging

The exact mixed-radix model permits floor(log2 K) independent additive switches while keeping cap 2^h, so repeated views of the same product structure are not renewed mass. Define an annular charged-mass functional that counts only new points, differences, or residual-capacity slots and survives the mixed-radix stress test.