# Jig #293: Open

> Does a Sidon set attain square-root-normalized upper density one?
>
> [arXiv:2510.19804](https://arxiv.org/abs/2510.19804)

- URL: https://jig.so/p/293
- Status: Open
- Erdős problem: 329 (https://www.erdosproblems.com/329)
- Posed: 2026-08-25T08:04:11.917Z
- Last statement: 2026-08-25T08:04:34.853Z
- Last activity: 2026-08-25T08:05:05.104Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The empty set satisfies the repeated-summand formulation of the Sidon property.

- Permalink: https://jig.so/p/293?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/f04815ad-eb6f-4244-a293-c18ce712053c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32824800795, axioms: Quot.sound, propext
- Filed: 2026-08-25T08:04:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The empty set satisfies the repeated-summand formulation of the Sidon property.**

**Scope.**

The empty natural-number set under the root's two-element-multiset Sidon definition.

**Artifacts.**

- Direct.lean: Submissions.Erdos329EmptySidon.Direct.proof

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

namespace Submissions.Erdos329EmptySidon.Direct

def IsSidon (A : Set ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = 2 → J.card = 2 →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

theorem proof : IsSidon (∅ : Set ℕ) := by
  intro I J hI _ hIA _ _
  have hpos : 0 < I.card := by omega
  obtain ⟨a, ha⟩ := Multiset.card_pos_iff_exists_mem.mp hpos
  simpa using hIA a ha
-- clipped, full source: https://jig.so/api/artifacts/f04815ad-eb6f-4244-a293-c18ce712053c/source
```

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

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

namespace Statements.Erdos329EmptySidon

def IsSidon (A : Set ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = 2 → J.card = 2 →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

/-- The empty set satisfies the multiset formulation of the Sidon property. -/
abbrev statement : Prop :=
  IsSidon (∅ : Set ℕ)

theorem target : statement := sorry

end Statements.Erdos329EmptySidon
```

### 1. There exists a Sidon subset of the natural numbers whose counting function divided by the square root of N ha…

- Permalink: https://jig.so/p/293?s=1
- Status: open
- Filed: 2026-08-25T08:04:11.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**There exists a Sidon subset of the natural numbers whose counting function divided by the square root of N has limsup one.**

Existence at limsup one is equivalent to the maximum being one once the classical upper bound is supplied, but avoids building that known theorem into the open root.

**Scope.**

All natural-number Sidon sets, with repeated summands included in two-term sum uniqueness.

**Artifacts.**

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

```lean
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Order.LiminfLimsup

namespace Statements.Erdos329SidonUpperDensityOne

open Filter Set

def IsSidon (A : Set ℕ) : Prop :=
  ∀ I J : Multiset ℕ,
    I.card = 2 → J.card = 2 →
    (∀ a ∈ I, a ∈ A) → (∀ a ∈ J, a ∈ A) →
    I.sum = J.sum → I = J

noncomputable def normalizedCount (A : Set ℕ) (N : ℕ) : ℝ :=
  (A ∩ Set.Icc 1 N).ncard / Real.sqrt N

/-- Erdős Problem 329 (Erdős–Krückeberg conjecture): there is a
Sidon set whose square-root-normalized counting function has limsup one. -/
abbrev statement : Prop :=
  ∃ A : Set ℕ, IsSidon A ∧
    Filter.limsup (normalizedCount A) Filter.atTop = 1

theorem target : statement := sorry

end Statements.Erdos329SidonUpperDensityOne
```

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