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

Reference: erdosproblems.com/40

open AdditiveCombinatorics Filter Real Setopen scoped Pointwisenamespace 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 $\limsup 1_A\ast 1_A(n)=\infty$?

Asked here in decision form: is there any such $g$ at all? Establishing the implication for even one $g(N) → \infty$ already answers Erdős Problem 28 positively, because a basis of order $2$ satisfies $\lvert A\cap {1,\ldots,N}\rvert \gg N^{1/2}$.

@[category research open, AMS 11] theorem erdos_40 : answer(sorry) ↔ ∃ g : ℕ → ℝ, Tendsto g atTop atTop ∧ Erdos40For g := ⊢ True ↔ ∃ g, Tendsto g atTop atTop ∧ Erdos40For g 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).

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 ↦ ↑(∑ p ∈ Finset.HasAntidiagonal.antidiagonal N, A.indicator (fun x ↦ 1) p.1 * A.indicator (fun x ↦ 1) p.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:n + 1 ≤ mthis✝:0 < mh_empty:¬Nonempty ↑(A ∩ Icc 1 m)this:m ∈ (A + A)ᶜ⊢ 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 ↦ ↑(∑ p ∈ Finset.HasAntidiagonal.antidiagonal N, A.indicator (fun x ↦ 1) p.1 * A.indicator (fun x ↦ 1) p.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:n + 1 ≤ mthis✝¹:0 < mh_empty:¬Nonempty ↑(A ∩ Icc 1 m)this✝:m ∈ (A + A)ᶜthis:m ≤ n⊢ 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 ↦ ↑(∑ p ∈ Finset.HasAntidiagonal.antidiagonal N, A.indicator (fun x ↦ 1) p.1 * A.indicator (fun x ↦ 1) p.2)) atTop = ⊤A:Set ℕhA:(A + A)ᶜ.Finiten:ℕhn:∀ i ∈ (A + A)ᶜ, i ≤ nm:ℕhm:n + 1 ≤ mthis:0 < m⊢ Finite ↑(A ∩ Icc 1 m) All goals completed! 🐙end Erdos40