1. FormalConjectures
  2. ErdosProblems
  3. «40»
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Erdős Problem 40

Reference: erdosproblems.com/40

open AdditiveCombinatorics Filter Real Setopen scoped Pointwise namespace Erdos40

The predicate for a function $g\colon\mathbb{N} → \mathbb{R})$ that $$\lvert A\cap {1,\ldots,N}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$.

def Erdos40For (g : ℕ → ℝ) : Prop := ∀ A : Set ℕ, (fun N : ℕ ↦ √N / g N) =O[atTop] (fun N ↦ ((A ∩ .Icc 1 N).ncard : ℝ)) → limsup (fun N ↦ (sumRep A N : ℕ∞)) atTop = ⊤

Given a set of functions $\mathbb{N} → \mathbb{R})$, we assert that for all $g$ in that set, if $g(N) → \infty$ then $$\lvert A\cap {1,\ldots,N}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$.

def Erdos40ForSet (G : Set (ℕ → ℝ)) : Prop := ∀ g ∈ G, Tendsto g atTop atTop → Erdos40For g

For what functions $g(N) → \infty$ is it true that $$\lvert A\cap {1,\ldots,N}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_40 : Erdos40ForSet answer(sorry) := ⊢ Erdos40ForSet sorry All goals completed! 🐙

If we don't pose additional conditions on the functions, then this is a stronger form of the Erdős-Turán conjecture, see Erdõs Problem 28, (since establishing this for any function $g(N) → \infty$ would imply a positive solution to Erdős Problem 28).

@[category textbook, AMS 11] theorem erdos_40.variants.implies_erdos_28 (h_erdos_40 : Erdos40ForSet .univ) : type_of% Erdos28.erdos_28 := h_erdos_40:Erdos40ForSet univ⊢ ∀ (A : Set ℕ), (A + A)ᶜ.Finite → limsup (fun n => ↑(sumRep A n)) atTop = ⊤ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤⊢ ∀ (A : Set ℕ), (A + A)ᶜ.Finite → limsup (fun n => ↑(sumRep A n)) atTop = ⊤ intro A h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ limsup (fun n => ↑(sumRep A n)) atTop = ⊤ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ Tendsto ?g atTop atToph_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ (fun N => √↑N / ?g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncardh_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ ℕ → ℝ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ ℕ → ℝh_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ Tendsto ?g atTop atToph_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ (fun N => √↑N / ?g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ ℕ → ℝ All goals completed! 🐙 h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ Tendsto ?g atTop atTop h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ Tendsto (fun N => (fun x => x ^ (1 / 2)) ↑N) atTop atTop All goals completed! 🐙 h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finite⊢ (fun N => √↑N / ?g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ n⊢ (fun N => √↑N / √↑N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ n⊢ ∀ᶠ (x : ℕ) in atTop, ‖√↑x / √↑x‖ ≤ 1 * ‖↑(A ∩ Icc 1 x).ncard‖ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ n⊢ ∃ a, ∀ b ≥ a, ‖√↑b / √↑b‖ ≤ 1 * ‖↑(A ∩ Icc 1 b).ncard‖ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ n⊢ ∀ b ≥ n + 1, ‖√↑b / √↑b‖ ≤ 1 * ‖↑(A ∩ Icc 1 b).ncard‖ intro m h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1⊢ ‖√↑m / √↑m‖ ≤ 1 * ‖↑(A ∩ Icc 1 m).ncard‖ have : 0 < m := h_erdos_40:Erdos40ForSet univ⊢ ∀ (A : Set ℕ), (A + A)ᶜ.Finite → limsup (fun n => ↑(sumRep A n)) atTop = ⊤ All goals completed! 🐙 h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ ‖1‖ ≤ ‖↑(A ∩ Icc 1 m).ncard‖ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ 1 ≤ (A ∩ Icc 1 m).ncard h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ Nonempty ↑(A ∩ Icc 1 m) ∧ Finite ↑(A ∩ Icc 1 m) h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ Nonempty ↑(A ∩ Icc 1 m)h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ Finite ↑(A ∩ Icc 1 m) h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ Nonempty ↑(A ∩ Icc 1 m) h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)⊢ False have : m ∈ (A + A)ᶜ := h_erdos_40:Erdos40ForSet univ⊢ ∀ (A : Set ℕ), (A + A)ᶜ.Finite → limsup (fun n => ↑(sumRep A n)) atTop = ⊤ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)h:m ∈ A + A⊢ False h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = m⊢ False h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = m⊢ Nonempty ↑(A ∩ Icc 1 m) h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':1 ≤ a⊢ Nonempty ↑(A ∩ Icc 1 m)h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':¬1 ≤ a⊢ Nonempty ↑(A ∩ Icc 1 m) h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':1 ≤ a⊢ Nonempty ↑(A ∩ Icc 1 m) refine ⟨a, ha, ha', h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':1 ≤ a⊢ a ≤ m All goals completed! 🐙⟩ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':¬1 ≤ a⊢ Nonempty ↑(A ∩ Icc 1 m) exact ⟨b, hb, h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':¬1 ≤ a⊢ 1 ≤ b h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:a + b = mha':¬1 ≤ a⊢ 1 ≤ b; All goals completed! 🐙, h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)a:ℕha:a ∈ Ab:ℕhb:b ∈ Ah:(fun x1 x2 => x1 + x2) a b = mha':¬1 ≤ a⊢ b ≤ m All goals completed! 🐙⟩ h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this✝¹:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm ah_empty:¬Nonempty ↑(A ∩ Icc 1 m)this✝:m ∈ (A + A)ᶜ := fun h => match h with | Exists.intro a ⟨ha, Exists.intro b ⟨hb, h⟩⟩ => absurd (if ha' : 1 ≤ a then Nonempty.intro ⟨a, ⟨ha, ⟨ha', Eq.ndrec (motive := fun m => m ≥ n + 1 → 0 < m → ¬Nonempty ↑(A ∩ Icc 1 m) → a ≤ m) (fun hm this h_empty => le_add_of_nonneg_right (of_eq_true (Eq.trans (one_le._simp_2 b) (eq_true True.intro)))) h hm this✝¹ h_empty⟩⟩⟩ else Nonempty.intro ⟨b, ⟨hb, ⟨Decidable.byContradiction fun a_1 => implies_erdos_28._proof_7 n m this✝¹ a b h ha' a_1, Eq.ndrec (motive := fun m => m ≥ n + 1 → 0 < m → ¬Nonempty ↑(A ∩ Icc 1 m) → b ≤ m) (fun hm this h_empty => le_add_of_nonneg_left (of_eq_true (Eq.trans (one_le._simp_2 a) (eq_true True.intro)))) h hm this✝¹ h_empty⟩⟩⟩) h_emptythis:m ≤ n := hn m this✝⊢ False All goals completed! 🐙 h_erdos_40:∀ (g : ℕ → ℝ), Tendsto g atTop atTop → ∀ (A : Set ℕ), ((fun N => √↑N / g N) =O[atTop] fun N => ↑(A ∩ Icc 1 N).ncard) → limsup (fun N => ↑(∑ x ∈ Finset.antidiagonal N, A.indicator (fun x => 1) x.1 * A.indicator (fun x => 1) x.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:m ≥ n + 1this:0 < m := Decidable.byContradiction fun a => implies_erdos_28._proof_4 n m hm a⊢ Finite ↑(A ∩ Icc 1 m) All goals completed! 🐙 end Erdos40