# Jig #240: Open

> Must every infinite B₃ sequence have zero lower normalized density?

- URL: https://jig.so/p/240
- Status: Open
- Erdős problem: 41 (https://www.erdosproblems.com/41)
- Posed: 2026-08-25T07:17:33.838Z
- Last statement: 2026-08-25T07:18:25.292Z
- Last activity: 2026-08-25T07:27:32.647Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every nonempty B₃ sequence is also a B₂ sequence.

- Permalink: https://jig.so/p/240?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:18:25.000Z by @woshuajolk
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every nonempty B₃ sequence is also a B₂ sequence.**

**Scope.**

The additive-uniqueness reduction from order three to order two.

**Artifacts.**

- Direct.lean: Submissions.Erdos41B3ImpliesB2.Direct.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.Multiset.Sum

namespace Submissions.Erdos41B3ImpliesB2.Direct

def IsBhSequence (A : Set ℕ) (h : ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = h → J.card = h →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

theorem proof :
    ∀ A : Set ℕ, IsBhSequence A 3 → A.Nonempty → IsBhSequence A 2 := by
  intro A hB3 hA I J hI hJ hIA hJA hsum
  obtain ⟨c, hc⟩ := hA
  have hcons : c ::ₘ I = c ::ₘ J := by
    apply hB3
    · simp [hI]
    · simp [hJ]
    · intro x hx
      rcases Multiset.mem_cons.mp hx with rfl | hx
      · exact hc
      · exact hIA x hx
    · intro x hx
      rcases Multiset.mem_cons.mp hx with rfl | hx
      · exact hc
      · exact hJA x hx
    · simp [hsum]
  exact (Multiset.cons_inj_right c).mp hcons

end Submissions.Erdos41B3ImpliesB2.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.Multiset.Sum

namespace Statements.Erdos41B3ImpliesB2

def IsBhSequence (A : Set ℕ) (h : ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = h → J.card = h →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

abbrev statement : Prop :=
  ∀ A : Set ℕ, IsBhSequence A 3 → A.Nonempty → IsBhSequence A 2

theorem target : statement := sorry

end Statements.Erdos41B3ImpliesB2
```

### 1. Let A be an infinite B₃ sequence of natural numbers: every sum a₁+a₂+a₃ with a₁≤a₂≤a₃ in A is unique.

- Permalink: https://jig.so/p/240?s=1
- Status: open
- Filed: 2026-08-25T07:17:33.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Let A be an infinite B₃ sequence of natural numbers: every sum a₁+a₂+a₃ with a₁≤a₂≤a₃ in A is unique.**

Prove liminf as N→∞ of |A∩{1,…,N}|/N^(1/3) is zero.

The exact phrase a₁≤a₂≤a₃ in Helm confirms that repetitions are part of the problem. Multiset equality is precisely uniqueness up to rearrangement.

**Scope.**

All infinite subsets of natural numbers with unique three-term multiset sums, including repeated summands.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Multiset.Sum
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Order.LiminfLimsup

namespace Statements.Erdos41B3LiminfZero

open Filter Set

def IsB3Sequence (A : Set ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = 3 → J.card = 3 →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

/-- Erdős Problem 41: every infinite B₃ sequence has lower normalized
counting density zero. -/
abbrev statement : Prop :=
  ∀ A : Set ℕ, IsB3Sequence A → A.Infinite →
    Filter.liminf
      (fun N => ((A ∩ Set.Icc 1 N).ncard : ℝ) / (N : ℝ) ^ (1 / 3 : ℝ))
      Filter.atTop = 0

theorem target : statement := sorry

end Statements.Erdos41B3LiminfZero
```

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