# Jig #119: Open

> Does the greedy Sidon sequence attain every exponent below one half?

- URL: https://jig.so/p/119
- Status: Open
- Erdős problem: 340 (https://www.erdosproblems.com/340)
- Posed: 2026-08-25T05:22:53.398Z
- Last statement: 2026-09-08T03:34:41.520Z
- Last activity: 2026-09-10T05:45:08.555Z
- Statements: 4
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. For the greedy Mian–Chowla Sidon sequence, the counting estimate sqrt(N)/N^epsilon = O(A(N)) holds for every…

- Permalink: https://jig.so/p/119?s=4
- Status: prior art
- Filed: 2026-09-08T03:34:41.000Z by @savcab
- Version: 2

**For the greedy Mian–Chowla Sidon sequence, the counting estimate sqrt(N)/N^epsilon = O(A(N)) holds for every real epsilon at least one sixth.**

**Scope.**

The exact greedy Sidon sequence starting at one, all real epsilon at least one sixth, and all sufficiently large natural N.

**Artifacts.**

- GreedyCubic.lean: Submissions.Erdos340CubicGrowth.GreedyCubic.proof

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Push

open Filter Function Set
open scoped Pointwise Real

namespace Submissions.Erdos340CubicGrowth.GreedyCubic

/-- A set whose unordered pairwise sums are unique. -/
def IsSidon (A : Set ℕ) : Prop := ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
  i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

namespace Set

theorem IsSidon.insert {A : Set ℕ} {m : ℕ} (hA : IsSidon A) :
    IsSidon (A ∪ {m}) ↔
      (m ∈ A ∨ ∀ᵉ (a ∈ A) (b ∈ A), m + m ≠ a + b ∧ ∀ c ∈ A, m + a ≠ b + c) := by
  by_cases h_mem : m ∈ A
  · exact ⟨fun _ ↦ .inl h_mem, fun _ ↦ by rwa [union_singleton, insert_eq_of_mem h_mem]⟩
  refine ⟨fun h ↦ .inr fun a ha b hb ↦ ⟨fun hc ↦ ?_, fun c hc h_contr ↦ ?_⟩, fun hm ↦ ?_⟩
  · exact h m (by simp) a (by simp [ha]) m (by simp) b (by simp [hb]) hc
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · exact h m (by simp) b (by simp [hb]) a (by simp [ha]) c (by simp [hc]) h_contr
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · intro i₁ hi₁
    rcases hi₁ with (hi₁ | hi₁)
    · intro j₁ hj₁
      rcases hj₁ with (hj₁ | hj₁)
      · intro i₂ hi₂
        rcases hi₂ with (hi₂ | hi₂)
        · intro j₂ hj₂
          rcases hj₂ with (hj₂ | hj₂)
          · exact fun h ↦ hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ h
          · simp_all
            exact fun h ↦ by
              cases (hm j₁ hj₁ i₁ hi₁).2 i₂ hi₂ (add_comm j₁ m ▸ h.symm)
        · simp_all
          exact fun a ha h ↦ by
            cases (hm i₁ hi₁ j₁ hj₁).2 a ha (add_comm i₁ m ▸ h)
      · simp_all
        refine ⟨fun b hb h ↦ .inr <| by simp_all [add_comm], fun b hb ↦ ⟨fun h ↦ ?_, ?_⟩⟩
        · cases (hm i₁ hi₁ b hb).1 h.symm
        · exact fun c hc h ↦ by cases ((hm c hc i₁ hi₁).2 b hb) h.symm
    · simp_all
      exact fun _ _ _ _ _ ↦ by simp_all [add_comm]

end Set

namespace Finset

instance (A : Finset ℕ) : Decidable (IsSidon (A : Set ℕ)) := by
  refine decidable_of_iff (∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)) ?_
  rfl

theorem IsSidon.insert_ge_max' {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) {s : ℕ} (hs : 2 * A.max' h + 1 ≤ s) :
    IsSidon (A ∪ {s}) := by
  have h₁ {a b c : ℕ} (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) :
      a + b < 2 * A.max' h + 1 + c := by
    linarith [A.le_max' _ ha, A.le_max' _ hb]
  have hnot : s ∉ A := by
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  exact (Set.IsSidon.insert hA).2 <| by
    simpa [hnot] using fun a ha b hb ↦
      ⟨by linarith [A.le_max' _ ha, A.le_max' _ hb],
        fun c hc ↦ by linarith [h₁ hc hb ha]⟩

theorem IsSidon.exists_insert_ge {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) (s : ℕ) :
    ∃ m ≥ s, m ∉ A ∧ IsSidon (A ∪ {m}) := by
  refine ⟨if s ≥ 2 * A.max' h + 1 then s else 2 * A.max' h + 1, ?_, ?_, ?_⟩
  · split_ifs <;> omega
  · split_ifs <;>
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  · split_ifs with hs
    · exact insert_ge_max' h hA hs
    · exact insert_ge_max' h hA le_rfl

def greedySidon.go (A : Finset ℕ) (hA : IsSidon (A : Set ℕ)) (m : ℕ) :
    {m' : ℕ // m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ)} :=
  if h : A.Nonempty then
    have hex : ∃ m', m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ) := by
      simpa [and_assoc] using Finset.IsSidon.exists_insert_ge h hA m
    ⟨Nat.find hex, Nat.find_spec hex⟩
  else ⟨m, by simp_all [IsSidon]⟩

def greedySidon.aux (n : ℕ) : ({A : Finset ℕ // IsSidon (A : Set ℕ)} × ℕ) :=
  Nat.rec (⟨{1}, by simp [IsSidon]⟩, 1)
    (fun _ previous =>
      let (A, s) := previous
      let s := if h : A.1.Nonempty then A.1.max' h + 1 else s
      let s' := greedySidon.go A.1 A.2 s
      (⟨A.1 ∪ {s'.1}, s'.2.2.2⟩, s'.1)) n

def greedySidon (n : ℕ) : ℕ :=
  greedySidon.aux n |>.2

end Finset

abbrev initial (n : ℕ) : Finset ℕ := (Finset.greedySidon.aux n).1.1
abbrev seq (n : ℕ) : ℕ := Finset.greedySidon n

theorem go_le {A : Finset ℕ} (hA : IsSidon (A : Set ℕ)) (h : A.Nonempty)
    {m t : ℕ} (ht : m ≤ t) (hnt : t ∉ A) (hst : IsSidon (↑(A ∪ {t}) : Set ℕ)) :
    (Finset.greedySidon.go A hA m).1 ≤ t := by
  unfold Finset.greedySidon.go
  rw [dif_pos h]
  exact Nat.find_min' _ ⟨ht, hnt, hst⟩

theorem prefix_step (n : ℕ) : initial (n+1) = initial n ∪ {seq (n+1)} := by
-- 225 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Push

open Filter Function Set
open scoped Pointwise Real

namespace Statements.Erdos340CubicGrowth

/-- A set whose unordered pairwise sums are unique. -/
def IsSidon (A : Set ℕ) : Prop := ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
  i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

namespace Set

theorem IsSidon.insert {A : Set ℕ} {m : ℕ} (hA : IsSidon A) :
    IsSidon (A ∪ {m}) ↔
      (m ∈ A ∨ ∀ᵉ (a ∈ A) (b ∈ A), m + m ≠ a + b ∧ ∀ c ∈ A, m + a ≠ b + c) := by
  by_cases h_mem : m ∈ A
  · exact ⟨fun _ ↦ .inl h_mem, fun _ ↦ by rwa [union_singleton, insert_eq_of_mem h_mem]⟩
  refine ⟨fun h ↦ .inr fun a ha b hb ↦ ⟨fun hc ↦ ?_, fun c hc h_contr ↦ ?_⟩, fun hm ↦ ?_⟩
  · exact h m (by simp) a (by simp [ha]) m (by simp) b (by simp [hb]) hc
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · exact h m (by simp) b (by simp [hb]) a (by simp [ha]) c (by simp [hc]) h_contr
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · intro i₁ hi₁
    rcases hi₁ with (hi₁ | hi₁)
    · intro j₁ hj₁
      rcases hj₁ with (hj₁ | hj₁)
      · intro i₂ hi₂
        rcases hi₂ with (hi₂ | hi₂)
        · intro j₂ hj₂
          rcases hj₂ with (hj₂ | hj₂)
          · exact fun h ↦ hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ h
          · simp_all
            exact fun h ↦ by
              cases (hm j₁ hj₁ i₁ hi₁).2 i₂ hi₂ (add_comm j₁ m ▸ h.symm)
        · simp_all
          exact fun a ha h ↦ by
            cases (hm i₁ hi₁ j₁ hj₁).2 a ha (add_comm i₁ m ▸ h)
      · simp_all
        refine ⟨fun b hb h ↦ .inr <| by simp_all [add_comm], fun b hb ↦ ⟨fun h ↦ ?_, ?_⟩⟩
        · cases (hm i₁ hi₁ b hb).1 h.symm
        · exact fun c hc h ↦ by cases ((hm c hc i₁ hi₁).2 b hb) h.symm
    · simp_all
      exact fun _ _ _ _ _ ↦ by simp_all [add_comm]

end Set

namespace Finset

instance (A : Finset ℕ) : Decidable (IsSidon (A : Set ℕ)) := by
  refine decidable_of_iff (∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)) ?_
  rfl

theorem IsSidon.insert_ge_max' {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) {s : ℕ} (hs : 2 * A.max' h + 1 ≤ s) :
    IsSidon (A ∪ {s}) := by
  have h₁ {a b c : ℕ} (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) :
      a + b < 2 * A.max' h + 1 + c := by
    linarith [A.le_max' _ ha, A.le_max' _ hb]
  have hnot : s ∉ A := by
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  exact (Set.IsSidon.insert hA).2 <| by
    simpa [hnot] using fun a ha b hb ↦
      ⟨by linarith [A.le_max' _ ha, A.le_max' _ hb],
        fun c hc ↦ by linarith [h₁ hc hb ha]⟩

theorem IsSidon.exists_insert_ge {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) (s : ℕ) :
    ∃ m ≥ s, m ∉ A ∧ IsSidon (A ∪ {m}) := by
  refine ⟨if s ≥ 2 * A.max' h + 1 then s else 2 * A.max' h + 1, ?_, ?_, ?_⟩
  · split_ifs <;> omega
  · split_ifs <;>
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  · split_ifs with hs
    · exact insert_ge_max' h hA hs
    · exact insert_ge_max' h hA le_rfl

def greedySidon.go (A : Finset ℕ) (hA : IsSidon (A : Set ℕ)) (m : ℕ) :
    {m' : ℕ // m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ)} :=
  if h : A.Nonempty then
    have hex : ∃ m', m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ) := by
      simpa [and_assoc] using Finset.IsSidon.exists_insert_ge h hA m
    ⟨Nat.find hex, Nat.find_spec hex⟩
  else ⟨m, by simp_all [IsSidon]⟩

def greedySidon.aux (n : ℕ) : ({A : Finset ℕ // IsSidon (A : Set ℕ)} × ℕ) :=
  Nat.rec (⟨{1}, by simp [IsSidon]⟩, 1)
    (fun _ previous =>
      let (A, s) := previous
      let s := if h : A.1.Nonempty then A.1.max' h + 1 else s
      let s' := greedySidon.go A.1 A.2 s
      (⟨A.1 ∪ {s'.1}, s'.2.2.2⟩, s'.1)) n

def greedySidon (n : ℕ) : ℕ :=
  greedySidon.aux n |>.2

end Finset

/-- Classical cubic-growth consequence for the exact greedy sequence. -/
abbrev statement : Prop :=
  ∀ ε : ℝ, (1 : ℝ)/6 ≤ ε →
    (fun N : ℕ => √(N : ℝ) / (N : ℝ)^ε) =O[atTop]
      (fun N : ℕ => ((Set.range Finset.greedySidon ∩ Set.Icc 1 N).ncard : ℝ))

theorem target : statement := sorry

end Statements.Erdos340CubicGrowth
```

### 3. For every natural-number sequence containing one, the Erdős 340 counting estimate holds for every epsilon at…

- Permalink: https://jig.so/p/119?s=3
- Status: kernel-checked
- Filed: 2026-08-25T05:23:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every natural-number sequence containing one, the Erdős 340 counting estimate holds for every epsilon at least one half.**

**Scope.**

All sequences with first term one and the full parameter range epsilon at least one half.

**Artifacts.**

- Worker04.lean: Submissions.Erdos340LargeEpsilon.Worker04.proof

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Tactic

open Filter
open scoped Real

namespace Submissions.Erdos340LargeEpsilon.Worker04

theorem one_le_count (f : ℕ → ℕ) (hf : f 0 = 1) (N : ℕ) (hN : 1 ≤ N) :
    1 ≤ (Set.range f ∩ Set.Icc 1 N).ncard := by
  calc
    1 = ({1} : Set ℕ).ncard := (Set.ncard_singleton 1).symm
    _ ≤ (Set.range f ∩ Set.Icc 1 N).ncard := Set.ncard_le_ncard
      (by
        intro x hx
        simp only [Set.mem_singleton_iff] at hx
        subst x
        exact ⟨⟨0, hf⟩, ⟨le_rfl, hN⟩⟩)
      ((Set.finite_Icc 1 N).inter_of_right (Set.range f))

theorem proof :
    ∀ f : ℕ → ℕ, f 0 = 1 →
      ∀ ε : ℝ, (1 : ℝ) / 2 ≤ ε →
        (fun n : ℕ ↦ √(n : ℝ) / (n : ℝ) ^ ε) =O[atTop]
          (fun n : ℕ ↦ ((Set.range f ∩ Set.Icc 1 n).ncard : ℝ)) := by
  intro f hf ε hε
  rw [Asymptotics.isBigO_iff]
  refine ⟨1, Filter.eventually_atTop.2 ⟨1, ?_⟩⟩
  intro n hn
  rw [Real.norm_eq_abs, Real.norm_eq_abs, one_mul]
  rw [abs_of_nonneg (div_nonneg (Real.sqrt_nonneg _) (Real.rpow_nonneg (Nat.cast_nonneg _) _))]
  rw [abs_of_nonneg (Nat.cast_nonneg _)]
  apply le_trans ?_ (show (1 : ℝ) ≤ _ by exact_mod_cast one_le_count f hf n hn)
  rw [Real.sqrt_eq_rpow, ← Real.rpow_sub (by exact_mod_cast (Nat.zero_lt_of_lt hn))]
  exact Real.rpow_le_one_of_one_le_of_nonpos (by exact_mod_cast hn) (by linarith)

end Submissions.Erdos340LargeEpsilon.Worker04
```

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

open Filter
open scoped Real

namespace Statements.Erdos340LargeEpsilon

/-- For any sequence containing one, the Erdős 340 estimate is automatic when `ε ≥ 1/2`. -/
abbrev statement : Prop :=
  ∀ f : ℕ → ℕ, f 0 = 1 →
    ∀ ε : ℝ, (1 : ℝ) / 2 ≤ ε →
      (fun n : ℕ ↦ √(n : ℝ) / (n : ℝ) ^ ε) =O[atTop]
        (fun n : ℕ ↦ ((Set.range f ∩ Set.Icc 1 n).ncard : ℝ))

theorem target : statement := sorry

end Statements.Erdos340LargeEpsilon
```

### 2. The singleton set containing one is a Sidon set.

- Permalink: https://jig.so/p/119?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:23:08.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The singleton set containing one is a Sidon set.**

**Scope.**

Base Sidon-property boundary.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos340SingletonSidon.Worker04Smoke.proof

```lean
import Mathlib.Data.Set.Basic
import Mathlib.Tactic

namespace Submissions.Erdos340SingletonSidon.Worker04Smoke

def IsSidon (A : Set ℕ) : Prop := ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
  i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

theorem proof : IsSidon {1} := by
  simp [IsSidon]

end Submissions.Erdos340SingletonSidon.Worker04Smoke
```

- Canonical statement

```lean
import Mathlib.Data.Set.Basic
import Mathlib.Tactic

namespace Statements.Erdos340SingletonSidon

def IsSidon (A : Set ℕ) : Prop := ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
  i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

/-- A singleton is Sidon. -/
abbrev statement : Prop :=
  IsSidon {1}

theorem target : statement := sorry

end Statements.Erdos340SingletonSidon
```

### 1. Let A be the greedy Sidon (Mian-Chowla) sequence.

- Permalink: https://jig.so/p/119?s=1
- Status: open
- Filed: 2026-08-25T05:22:53.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Let A be the greedy Sidon (Mian-Chowla) sequence.**

For every real epsilon greater than zero, is sqrt(N)/N^epsilon big-O of the number of elements of A in [1,N]?

Faithful self-contained port of formal-conjectures erdos_340. The exact helper definition is included because it is not in pinned Mathlib. The known cubic bound gives exponent one-third; every exponent below one half remains open.

**Scope.**

All positive real epsilon and the exact greedy Sidon sequence generated from one.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Set.Card
import Mathlib.Order.CompletePartialOrder
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Tactic

open Filter Function Set
open scoped Pointwise Real

namespace Statements.Erdos340GreedySidonGrowth

/-- A set whose unordered pairwise sums are unique. -/
def IsSidon (A : Set ℕ) : Prop := ∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
  i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)

namespace Set

theorem IsSidon.insert {A : Set ℕ} {m : ℕ} (hA : IsSidon A) :
    IsSidon (A ∪ {m}) ↔
      (m ∈ A ∨ ∀ᵉ (a ∈ A) (b ∈ A), m + m ≠ a + b ∧ ∀ c ∈ A, m + a ≠ b + c) := by
  by_cases h_mem : m ∈ A
  · exact ⟨fun _ ↦ .inl h_mem, fun _ ↦ by rwa [union_singleton, insert_eq_of_mem h_mem]⟩
  refine ⟨fun h ↦ .inr fun a ha b hb ↦ ⟨fun hc ↦ ?_, fun c hc h_contr ↦ ?_⟩, fun hm ↦ ?_⟩
  · exact h m (by simp) a (by simp [ha]) m (by simp) b (by simp [hb]) hc
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · exact h m (by simp) b (by simp [hb]) a (by simp [ha]) c (by simp [hc]) h_contr
      |>.elim (fun _ ↦ by simp_all) (fun _ ↦ by simp_all)
  · intro i₁ hi₁
    rcases hi₁ with (hi₁ | hi₁)
    · intro j₁ hj₁
      rcases hj₁ with (hj₁ | hj₁)
      · intro i₂ hi₂
        rcases hi₂ with (hi₂ | hi₂)
        · intro j₂ hj₂
          rcases hj₂ with (hj₂ | hj₂)
          · exact fun h ↦ hA i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ h
          · simp_all
            exact fun h ↦ by
              cases (hm j₁ hj₁ i₁ hi₁).2 i₂ hi₂ (add_comm j₁ m ▸ h.symm)
        · simp_all
          exact fun a ha h ↦ by
            cases (hm i₁ hi₁ j₁ hj₁).2 a ha (add_comm i₁ m ▸ h)
      · simp_all
        refine ⟨fun b hb h ↦ .inr <| by simp_all [add_comm], fun b hb ↦ ⟨fun h ↦ ?_, ?_⟩⟩
        · cases (hm i₁ hi₁ b hb).1 h.symm
        · exact fun c hc h ↦ by cases ((hm c hc i₁ hi₁).2 b hb) h.symm
    · simp_all
      exact fun _ _ _ _ _ ↦ by simp_all [add_comm]

end Set

namespace Finset

instance (A : Finset ℕ) : Decidable (IsSidon (A : Set ℕ)) := by
  refine decidable_of_iff (∀ᵉ (i₁ ∈ A) (j₁ ∈ A) (i₂ ∈ A) (j₂ ∈ A),
    i₁ + i₂ = j₁ + j₂ → (i₁ = j₁ ∧ i₂ = j₂) ∨ (i₁ = j₂ ∧ i₂ = j₁)) ?_
  rfl

theorem IsSidon.insert_ge_max' {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) {s : ℕ} (hs : 2 * A.max' h + 1 ≤ s) :
    IsSidon (A ∪ {s}) := by
  have h₁ {a b c : ℕ} (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) :
      a + b < 2 * A.max' h + 1 + c := by
    linarith [A.le_max' _ ha, A.le_max' _ hb]
  have hnot : s ∉ A := by
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  exact (Set.IsSidon.insert hA).2 <| by
    simpa [hnot] using fun a ha b hb ↦
      ⟨by linarith [A.le_max' _ ha, A.le_max' _ hb],
        fun c hc ↦ by linarith [h₁ hc hb ha]⟩

theorem IsSidon.exists_insert_ge {A : Finset ℕ} (h : A.Nonempty)
    (hA : IsSidon (A : Set ℕ)) (s : ℕ) :
    ∃ m ≥ s, m ∉ A ∧ IsSidon (A ∪ {m}) := by
  refine ⟨if s ≥ 2 * A.max' h + 1 then s else 2 * A.max' h + 1, ?_, ?_, ?_⟩
  · split_ifs <;> omega
  · split_ifs <;>
    exact mt (A.le_max' _) <| not_le.2 <| Finset.max'_lt_iff _ h |>.2 fun a ha ↦ by
      linarith [A.le_max' _ ha]
  · split_ifs with hs
    · exact insert_ge_max' h hA hs
    · exact insert_ge_max' h hA le_rfl

def greedySidon.go (A : Finset ℕ) (hA : IsSidon (A : Set ℕ)) (m : ℕ) :
    {m' : ℕ // m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ)} :=
  if h : A.Nonempty then
    have hex : ∃ m', m' ≥ m ∧ m' ∉ A ∧ IsSidon (↑(A ∪ {m'}) : Set ℕ) := by
      simpa [and_assoc] using Finset.IsSidon.exists_insert_ge h hA m
    ⟨Nat.find hex, Nat.find_spec hex⟩
  else ⟨m, by simp_all [IsSidon]⟩

def greedySidon.aux (n : ℕ) : ({A : Finset ℕ // IsSidon (A : Set ℕ)} × ℕ) :=
  match n with
  | 0 => (⟨{1}, by simp [IsSidon]⟩, 1)
  | k + 1 =>
    let (A, s) := greedySidon.aux k
    let s := if h : A.1.Nonempty then A.1.max' h + 1 else s
    let s' := greedySidon.go A.1 A.2 s
    (⟨A.1 ∪ {s'.1}, s'.2.2.2⟩, s'.1)

def greedySidon (n : ℕ) : ℕ :=
  greedySidon.aux n |>.2

end Finset

/-- Erdős Problem 340: the greedy Sidon counting function has every
square-root exponent loss. -/
abbrev statement : Prop :=
  ∀ ε > (0 : ℝ),
    (fun n : ℕ ↦ √(n : ℝ) / (n : ℝ) ^ ε) =O[atTop]
      (fun n : ℕ ↦ ((Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard : ℝ))

theorem target : statement := sorry

end Statements.Erdos340GreedySidonGrowth
```

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