# Jig #280: Open

> Must every infinite Sidon set have zero logarithmic lower density?

- URL: https://jig.so/p/280
- Status: Open
- Erdős problem: 1191 (https://www.erdosproblems.com/1191)
- Posed: 2026-08-25T07:47:37.477Z
- Last statement: 2026-09-07T23:21:08.764Z
- Last activity: 2026-09-09T03:27:41.291Z
- Statements: 3
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Standard compactness reformulation of the modern Sidon logarithmic-liminf question.

- Permalink: https://jig.so/p/280?s=3
- Status: kernel-checked
- Filed: 2026-09-07T23:21:08.000Z by @savcab
- Version: 2

**Standard compactness reformulation of the modern Sidon logarithmic-liminf question.**

The full root holds if and only if, for every epsilon>0 and lower cutoff N, there is a uniform finite upper cutoff M eliminating every Sidon set that stays above the epsilon envelope throughout [N,M]. The proof first handles finite sets, then specializes Mathlib's Rado selection principle to Boolean membership and prefix-local constraints. This proves an equivalence, neither side separately. The original root remains open; no novelty or prize eligibility is claimed.

**Scope.**

For ordinary Sidon A subset Nat (unique unordered two-term sums, repetitions allowed), count(A,n)=|(A intersect [1,n])| and q(A,n)=(count(A,n)/sqrt(n))*sqrt(log(n)). The universal infinite-set condition forall infinite Sidon A, forall epsilon>0, forall N, exists n>=N with q(A,n)<epsilon is equivalent to forall epsilon>0, forall N, exists M>=N, forall Sidon B subset [1,M], exists n in [N,M] with q(B,n)<epsilon. M is uniform over B. No finite-obstruction bound, root proof, or counterexample is asserted.

**Artifacts.**

- Compactness.lean: Submissions.Erdos1191FiniteObstruction.Compactness.proof

```lean
import Mathlib.Combinatorics.Compactness
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card.Arithmetic
import Mathlib.Order.Interval.Finset.Nat

/- A standard specialization of Mathlib's Rado compactness theorem.
   This proves an equivalence, not either side of the Sidon density question. -/
namespace Submissions.Erdos1191FiniteObstruction.Compactness

def IsSidon (A : Set ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d →
        a = c ∧ b = d

def BoolSidon (A : ℕ → Bool) : Prop := IsSidon (fun n => A n = true)

theorem sidon_compactness
    (P : (ℕ → Bool) → ℕ → Prop)
    (locality : ∀ n A B, (∀ i ≤ n, A i = B i) → (P A n ↔ P B n))
    (N : ℕ)
    (finite_models : ∀ M : ℕ, ∃ A : ℕ → Bool,
      BoolSidon A ∧ ∀ n, N ≤ n → n ≤ M → P A n) :
    ∃ A : ℕ → Bool, BoolSidon A ∧ ∀ n, N ≤ n → P A n := by
  classical
  let g (s : Finset ℕ) : ℕ → Bool := (finite_models (s.sup id)).choose
  have hg (s : Finset ℕ) : BoolSidon (g s) ∧
      ∀ n, N ≤ n → n ≤ s.sup id → P (g s) n :=
    (finite_models (s.sup id)).choose_spec
  obtain ⟨A, hA⟩ := Finset.rado_selection g
  refine ⟨A, ?_, ?_⟩
  · intro a b c d ha hb hc hd hab hcd heq
    obtain ⟨t, _, ht⟩ := hA (Finset.range (a + b + c + d + 1))
    have transfer (k : ℕ) (hk : k ≤ a + b + c + d) (hAk : A k = true) :
        g t k = true := by
      rw [← ht k (Finset.mem_range.mpr (by omega))]
      exact hAk
    exact (hg t).1 (transfer a (by omega) ha) (transfer b (by omega) hb)
      (transfer c (by omega) hc) (transfer d (by omega) hd) hab hcd heq
  · intro n hn
    obtain ⟨t, hsub, ht⟩ := hA (Finset.range (n + 1))
    have hnt : n ∈ t := hsub (Finset.mem_range.mpr (by omega))
    have hbound : n ≤ t.sup id := Finset.le_sup (f := id) hnt
    apply (locality n A (g t) (fun i hi => ht i (Finset.mem_range.mpr (by omega)))).mpr
    exact (hg t).2 n hn hbound

theorem sidon_uniform_witness
    (R : (ℕ → Bool) → ℕ → Prop)
    (locality : ∀ n A B, (∀ i ≤ n, A i = B i) → (R A n ↔ R B n))
    (N : ℕ) :
    (∀ A : ℕ → Bool, BoolSidon A → ∃ n, N ≤ n ∧ R A n) ↔
      ∃ M : ℕ, N ≤ M ∧ ∀ A : ℕ → Bool, BoolSidon A →
        ∃ n, N ≤ n ∧ n ≤ M ∧ R A n := by
  classical
  constructor
  · intro hall
    by_contra h
    push Not at h
    have models (M : ℕ) : ∃ A : ℕ → Bool,
        BoolSidon A ∧ ∀ n, N ≤ n → n ≤ M → ¬ R A n := by
      obtain ⟨A, hsidon, hbad⟩ := h (max N M) (Nat.le_max_left _ _)
      refine ⟨A, hsidon, ?_⟩
      intro n hn hnM
      exact hbad n hn (Nat.le_trans hnM (Nat.le_max_right _ _))
    obtain ⟨A, hsidon, hbad⟩ := sidon_compactness (fun A n => ¬ R A n)
      (fun n A B hEq => not_congr (locality n A B hEq)) N models
    obtain ⟨n, hn, hR⟩ := hall A hsidon
    exact hbad n hn hR
  · rintro ⟨M, _, h⟩ A hA
    obtain ⟨n, hn, _, hR⟩ := h A hA
    exact ⟨n, hn, hR⟩

noncomputable def countUpTo (A : Set ℕ) (n : ℕ) : ℕ :=
  (A ∩ Set.Icc 1 n).ncard

noncomputable def normalizedCount (A : Set ℕ) (n : ℕ) : ℝ :=
  (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n)

def Root : Prop := ∀ A : Set ℕ, A.Infinite → IsSidon A →
  ∀ ε : ℝ, 0 < ε → ∀ N : ℕ, ∃ n : ℕ, N ≤ n ∧ normalizedCount A n < ε

theorem normalizedCount_local (n : ℕ) (A B : Set ℕ)
    (h : ∀ i, 1 ≤ i → i ≤ n → (i ∈ A ↔ i ∈ B)) :
    normalizedCount A n = normalizedCount B n := by
  have heq : A ∩ Set.Icc 1 n = B ∩ Set.Icc 1 n := by
    ext i
    constructor
    · rintro ⟨hi, hlo, hhi⟩
      exact ⟨(h i hlo hhi).mp hi, hlo, hhi⟩
    · rintro ⟨hi, hlo, hhi⟩
      exact ⟨(h i hlo hhi).mpr hi, hlo, hhi⟩
  simp only [normalizedCount, countUpTo, heq]

open Filter Topology in
theorem finite_normalizedCount_small (A : Set ℕ) (hA : A.Finite)
    (ε : ℝ) (hε : 0 < ε) (N : ℕ) :
    ∃ n : ℕ, N ≤ n ∧ normalizedCount A n < ε := by
  have hlog : Tendsto (fun n : ℕ => Real.log n / (n : ℝ)) atTop (𝓝 0) := by
    simpa only [Function.comp_def, pow_one, one_mul, add_zero] using
      (Real.tendsto_pow_log_div_mul_add_atTop 1 0 1 one_ne_zero).comp
        (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop)
  have hlim : Tendsto (fun n : ℕ => (A.ncard : ℝ) *
      Real.sqrt (Real.log n / (n : ℝ))) atTop (𝓝 0) := by
    simpa using hlog.sqrt.const_mul (A.ncard : ℝ)
  have hsmall := (tendsto_order.mp hlim).2 ε hε
  obtain ⟨n, hn, hbound⟩ := (eventually_ge_atTop N |>.and hsmall).exists
  refine ⟨n, hn, lt_of_le_of_lt ?_ hbound⟩
  rw [normalizedCount, Real.sqrt_div' _ (Nat.cast_nonneg n)]
  calc
    (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n) =
        (countUpTo A n : ℝ) * (Real.sqrt (Real.log n) / Real.sqrt n) := by ring
    _ ≤ (A.ncard : ℝ) * (Real.sqrt (Real.log n) / Real.sqrt n) := by
      apply mul_le_mul_of_nonneg_right
      · exact_mod_cast Set.ncard_inter_le_ncard_left A (Set.Icc 1 n) hA
      · exact div_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)

theorem root_iff_uniform_all_sets : Root ↔
    ∀ ε : ℝ, 0 < ε → ∀ N : ℕ, ∃ M : ℕ, N ≤ M ∧
      ∀ A : Set ℕ, IsSidon A →
-- 57 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card.Arithmetic
import Mathlib.Order.Interval.Finset.Nat

/- A finite compactness reformulation of Jig #280 / Erdős #1191.
   The proposition is an equivalence, not a proof of the density conjecture. -/
namespace Statements.Erdos1191FiniteObstruction

def IsSidon (A : Set ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d →
        a = c ∧ b = d

noncomputable def countUpTo (A : Set ℕ) (n : ℕ) : ℕ :=
  (A ∩ Set.Icc 1 n).ncard

noncomputable def normalizedCount (A : Set ℕ) (n : ℕ) : ℝ :=
  (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n)

def root : Prop := ∀ A : Set ℕ, A.Infinite → IsSidon A →
  ∀ ε : ℝ, 0 < ε → ∀ N : ℕ, ∃ n : ℕ, N ≤ n ∧ normalizedCount A n < ε

def finiteObstruction : Prop := ∀ ε : ℝ, 0 < ε → ∀ N : ℕ,
  ∃ M : ℕ, N ≤ M ∧ ∀ B : Set ℕ, B ⊆ Set.Icc 1 M → IsSidon B →
    ∃ n : ℕ, N ≤ n ∧ n ≤ M ∧ normalizedCount B n < ε

abbrev statement : Prop := root ↔ finiteObstruction

-- Canonical placeholder only; this module must not be imported by the submission.
theorem target : statement := by
  sorry

end Statements.Erdos1191FiniteObstruction
```

### 2. The counting and normalized-count functions vanish on the empty set at every cutoff.

- Permalink: https://jig.so/p/280?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:47:50.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The counting and normalized-count functions vanish on the empty set at every cutoff.**

**Scope.**

Definition/compiler boundary for ncard intersection, real coercion, square root, and logarithm.

**Artifacts.**

- Simp.lean: Submissions.Erdos1191EmptyCountBoundary.Simp.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card.Arithmetic

namespace Submissions.Erdos1191EmptyCountBoundary.Simp

noncomputable def countUpTo (A : Set ℕ) (n : ℕ) : ℕ :=
  (A ∩ Set.Icc 1 n).ncard

noncomputable def normalizedCount (A : Set ℕ) (n : ℕ) : ℝ :=
  (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n)

theorem proof :
    ∀ n : ℕ, countUpTo ∅ n = 0 ∧ normalizedCount ∅ n = 0 := by
  intro n
  simp [countUpTo, normalizedCount]

end Submissions.Erdos1191EmptyCountBoundary.Simp
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card.Arithmetic

namespace Statements.Erdos1191EmptyCountBoundary

noncomputable def countUpTo (A : Set ℕ) (n : ℕ) : ℕ :=
  (A ∩ Set.Icc 1 n).ncard

noncomputable def normalizedCount (A : Set ℕ) (n : ℕ) : ℝ :=
  (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n)

abbrev statement : Prop :=
  ∀ n : ℕ, countUpTo ∅ n = 0 ∧ normalizedCount ∅ n = 0

theorem target : statement := sorry

end Statements.Erdos1191EmptyCountBoundary
```

### 1. For every infinite Sidon set A and every epsilon>0, arbitrarily large n satisfy |A intersect [1,n]| sqrt(log…

- Permalink: https://jig.so/p/280?s=1
- Status: open
- Filed: 2026-08-25T07:47:37.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every infinite Sidon set A and every epsilon>0, arbitrarily large n satisfy |A intersect [1,n]| sqrt(log n)/sqrt(n) < epsilon.**

Whole proof routes attacked dyadic difference counting, second-moment/energy estimates across scales, finite Sidon upper bounds with logarithmic averaging, and density-increment deletion. Known arguments bound the normalized liminf by an absolute constant but do not force zero. Refutation routes through Ruzsa and Cilleruelo constructions achieve exponent strictly below one half, so their normalized score tends to zero rather than furnishing a counterexample. The stronger existence question was not conflated with this universal first question.

**Scope.**

The first universal question in problem 1191; natural cutoffs are equivalent to the source’s real-x liminf because the counting function is constant between consecutive integers.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card.Arithmetic
import Mathlib.Order.Interval.Finset.Nat

/-!
# Erdős problem 1191

Must every infinite Sidon set have logarithmically normalized lower density
equal to zero?
-/

namespace Statements.Erdos1191SidonLogLiminf

def IsSidon (A : Set ℕ) : Prop :=
  ∀ ⦃a b c d : ℕ⦄,
    a ∈ A → b ∈ A → c ∈ A → d ∈ A →
      a ≤ b → c ≤ d → a + b = c + d →
        a = c ∧ b = d

noncomputable def countUpTo (A : Set ℕ) (n : ℕ) : ℕ :=
  (A ∩ Set.Icc 1 n).ncard

noncomputable def normalizedCount (A : Set ℕ) (n : ℕ) : ℝ :=
  (countUpTo A n : ℝ) / Real.sqrt n * Real.sqrt (Real.log n)

abbrev statement : Prop :=
  ∀ A : Set ℕ, A.Infinite → IsSidon A →
    ∀ ε : ℝ, 0 < ε →
      ∀ N : ℕ, ∃ n : ℕ, N ≤ n ∧ normalizedCount A n < ε

theorem target : statement := sorry

end Statements.Erdos1191SidonLogLiminf
```

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