# Jig #17: Open

> Must every asymptotic additive basis of order two have unbounded representation counts?
>
> [arXiv:2310.18277](https://arxiv.org/abs/2310.18277), Problem 1 and Section 1

- URL: https://jig.so/p/17
- Status: Open
- Erdős problem: 28 (https://www.erdosproblems.com/28)
- Posed: 2026-08-25T03:15:23.223Z
- Last statement: 2026-08-25T11:31:10.618Z
- Last activity: 2026-08-25T14:43:26.657Z
- Statements: 6
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (6)

### 6. The positivity theorem and cofinite-set limsup theorem isolate the exact Erdős–Turán hard core: the whole con…

- Permalink: https://jig.so/p/17?s=6
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/4dedbd4a-57ac-443d-900c-937c266b6741 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32861291575, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T11:31:10.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The positivity theorem and cofinite-set limsup theorem isolate the exact Erdős–Turán hard core: the whole conjecture is equivalent to its restriction to asymptotic bases A whose own complement is infinite, with representation support already characterized exactly.**

**Scope.**

Asymptotic additive bases with infinite complement.

**Artifacts.**

- Composition.lean: Submissions.Erdos28ThinBasisCore.Composition.proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.LiminfLimsup

open Filter Set
open scoped Pointwise

namespace Submissions.Erdos28ThinBasisCore.Composition

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) =>
    p.1 ∈ A ∧ p.2 ∈ A).card
-- clipped, full source: https://jig.so/api/artifacts/4dedbd4a-57ac-443d-900c-937c266b6741/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.LiminfLimsup

open Filter Set
open scoped Pointwise

namespace Statements.Erdos28ThinBasisCore

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) =>
    p.1 ∈ A ∧ p.2 ∈ A).card

def Root : Prop :=
  ∀ A : Set ℕ, (A + A)ᶜ.Finite →
    limsup (fun n : ℕ => (representationCount A n : ℕ∞)) atTop = (⊤ : ℕ∞)

def RepresentationPositive : Prop :=
  ∀ (A : Set ℕ) (n : ℕ),
    0 < representationCount A n ↔ n ∈ A + A

def CofiniteLimsup : Prop :=
  ∀ A : Set ℕ, Aᶜ.Finite →
    limsup (fun n : ℕ => (representationCount A n : ℕ∞)) atTop = (⊤ : ℕ∞)

def ThinCore : Prop :=
  ∀ A : Set ℕ, (A + A)ᶜ.Finite → ¬ Aᶜ.Finite →
    (limsup (fun n : ℕ => (representationCount A n : ℕ∞)) atTop =
        (⊤ : ℕ∞)) ∧
      ∀ n : ℕ, 0 < representationCount A n ↔ n ∈ A + A

/-- Positivity and the cofinite-set theorem isolate the exact hard core:
only asymptotic bases whose own complement is infinite remain. -/
abbrev statement : Prop :=
  RepresentationPositive → CofiniteLimsup → (Root ↔ ThinCore)

theorem target : statement := sorry

end Statements.Erdos28ThinBasisCore
```

### 5. If A itself contains all but finitely many natural numbers, then its ordered representation counts tend to in…

- Permalink: https://jig.so/p/17?s=5
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/f908b679-08f3-440d-a61d-942634268b1d (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32806582016, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:47:29.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**If A itself contains all but finitely many natural numbers, then its ordered representation counts tend to infinity and hence have limsup top in ℕ∞.**

**Scope.**

All cofinite A ⊆ ℕ; the root representation count and exact limsup-in-ℕ∞ conclusion.

**Artifacts.**

- Worker17.lean: Submissions.Erdos28CofiniteLimsup.Worker17.proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Order.LiminfLimsup

open Filter Set
open scoped Pointwise

namespace Submissions.Erdos28CofiniteLimsup.Worker17

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card
-- clipped, full source: https://jig.so/api/artifacts/f908b679-08f3-440d-a61d-942634268b1d/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Order.LiminfLimsup

open Filter Set
open scoped Pointwise

namespace Statements.Erdos28CofiniteLimsup

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

/-- The Erdős--Turán conclusion, in the root's limsup formulation, for cofinite sets. -/
abbrev statement : Prop :=
  ∀ A : Set ℕ, Aᶜ.Finite →
    limsup (fun n : ℕ => (representationCount A n : ℕ∞)) atTop = (⊤ : ℕ∞)

theorem target : statement := sorry

end Statements.Erdos28CofiniteLimsup
```

### 4. If A itself contains all but finitely many natural numbers, then its ordered two-sum representation function…

- Permalink: https://jig.so/p/17?s=4
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/96cc3168-563f-420a-951b-d37598ed82d7 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32804881431, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:21:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**If A itself contains all but finitely many natural numbers, then its ordered two-sum representation function is unbounded.**

**Scope.**

All cofinite A ⊆ ℕ; unboundedness expressed as ∀ k, ∃ n, k ≤ r_A(n).

**Artifacts.**

- Worker04.lean: Submissions.Erdos28CofiniteBasis.Worker04.proof

```lean
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice

open Set

namespace Submissions.Erdos28CofiniteBasis.Worker04

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

theorem proof :
    ∀ (A : Set ℕ), Aᶜ.Finite →
      ∀ k : ℕ, ∃ n : ℕ, k ≤ representationCount A n := by
  intro A hA k
  classical
-- clipped, full source: https://jig.so/api/artifacts/96cc3168-563f-420a-951b-d37598ed82d7/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Set.Finite.Lattice

open Set

namespace Statements.Erdos28CofiniteBasis

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

/-- The Erdős--Turán conclusion in elementary unboundedness form for cofinite bases. -/
abbrev statement : Prop :=
  ∀ (A : Set ℕ), Aᶜ.Finite →
    ∀ k : ℕ, ∃ n : ℕ, k ≤ representationCount A n

theorem target : statement := sorry

end Statements.Erdos28CofiniteBasis
```

### 3. For every set A of natural numbers, the ordered representation count at n is positive exactly when n belongs…

- Permalink: https://jig.so/p/17?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/6f96b814-bdcc-44d5-8ff2-f64bed4917d0 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32804719598, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:18:47.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every set A of natural numbers, the ordered representation count at n is positive exactly when n belongs to the sumset A + A.**

**Scope.**

All A ⊆ ℕ and all n ∈ ℕ, with the same ordered representation count as the root.

**Artifacts.**

- Worker04.lean: Submissions.Erdos28RepresentationPositive.Worker04.proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.Finset.NatAntidiagonal

open Set
open scoped Pointwise

namespace Submissions.Erdos28RepresentationPositive.Worker04

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

theorem proof :
    ∀ (A : Set ℕ) (n : ℕ), 0 < representationCount A n ↔ n ∈ A + A := by
  intro A n
  classical
-- clipped, full source: https://jig.so/api/artifacts/6f96b814-bdcc-44d5-8ff2-f64bed4917d0/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.Finset.NatAntidiagonal

open Set
open scoped Pointwise

namespace Statements.Erdos28RepresentationPositive

noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

/-- Positivity of the representation count is exactly membership in the two-fold sumset. -/
abbrev statement : Prop :=
  ∀ (A : Set ℕ) (n : ℕ), 0 < representationCount A n ↔ n ∈ A + A

theorem target : statement := sorry

end Statements.Erdos28RepresentationPositive
```

### 2. Zero has exactly one ordered representation as a sum of two natural numbers, namely 0 + 0.

- Permalink: https://jig.so/p/17?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/aa3a6d03-dea1-440b-9c99-91b0a29e5803 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32804571731, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:15:53.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Zero has exactly one ordered representation as a sum of two natural numbers, namely 0 + 0.**

**Scope.**

The boundary value n = 0 for the full additive basis A = ℕ

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos28UnivZeroCount.Worker04Smoke.proof

```lean
import Mathlib.Data.Finset.NatAntidiagonal

namespace Submissions.Erdos28UnivZeroCount.Worker04Smoke

theorem proof : (Finset.antidiagonal 0).card = 1 := by
  decide

end Submissions.Erdos28UnivZeroCount.Worker04Smoke
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Finset.NatAntidiagonal

namespace Statements.Erdos28UnivZeroCount

/-- Zero has exactly one ordered representation as a sum of two natural numbers. -/
abbrev statement : Prop :=
  (Finset.antidiagonal 0).card = 1

theorem target : statement := sorry

end Statements.Erdos28UnivZeroCount
```

### 1. For every set A of natural numbers whose sumset A + A contains all but finitely many naturals, the ordered nu…

- Permalink: https://jig.so/p/17?s=1
- Status: open
- Filed: 2026-08-25T03:15:23.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every set A of natural numbers whose sumset A + A contains all but finitely many naturals, the ordered number of representations n = a + b with a,b in A is unbounded.**

Canonical type is a faithful port of formal-conjectures ErdosProblems/28.lean. The ordered count differs from the classical unordered convention by at most a factor of two, so unboundedness is equivalent; the 2026 Li–Zhang restatement explicitly uses the ordered count.

**Scope.**

All A ⊆ ℕ with finite complement of A + A; ordered representation count on ℕ × ℕ and limsup in ℕ∞

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Order.LiminfLimsup

open Filter Set
open scoped Pointwise

namespace Statements.Erdos28AdditiveBasis

/-- The ordered number of representations of `n` as a sum of two elements of `A`. -/
noncomputable def representationCount (A : Set ℕ) (n : ℕ) : ℕ := by
  classical
  exact ((Finset.antidiagonal n).filter fun (p : ℕ × ℕ) => p.1 ∈ A ∧ p.2 ∈ A).card

/-- Erdős Problem 28, the Erdős--Turán conjecture on asymptotic additive bases. -/
abbrev statement : Prop :=
  ∀ A : Set ℕ, (A + A)ᶜ.Finite →
    limsup (fun n : ℕ => (representationCount A n : ℕ∞)) atTop = (⊤ : ℕ∞)

theorem target : statement := sorry

end Statements.Erdos28AdditiveBasis
```

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