FormalConjectures.ErdosProblems.«330»

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import FormalConjecturesUtil

Erdős Problem 330

Reference: erdosproblems.com/330


namespace Erdos330


open Set

open scoped BigOperators

Rep A m h means m is a sum of at most h elements of Ax.

def Rep (A : Set ℕ) (m h : ℕ) : Prop :=
  ∃ k : ℕ, k ≤ h ∧ ∃ f : Fin k → ℕ, (∀ i, f i ∈ A) ∧ (∑ i : Fin k, f i) = m

Integers not representable as a finite sum of elements with at most h terms of A while avoiding n.

def UnrepWithout (A : Set ℕ) (n h: ℕ) : Set ℕ :=
  {m | ¬ Rep (A \ {n}) m h}

An asymptotic additive basis of order h is minimal when one cannot obtain an asymptotic additive basis by removing any element from it.

def MinAsymptoticAddBasisOfOrder (A : Set ℕ) (h : ℕ) : Prop :=
  IsAsymptoticAddBasisOfOrder A h ∧ ∀ n ∈ A, ¬ IsAsymptoticAddBasisOfOrder (A \ {n}) h

Does there exist a minimal basis $A \subset \mathbb{N}$ with positive density such that, for any $n \in A$, the (upper) density of integers which cannot be represented without using $n$ is positive?

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'erdos_330_statement :
    answer(sorry) ↔ ∃ (A : Set ℕ), ∃ h, MinAsymptoticAddBasisOfOrder A h ∧ A.HasPosDensity ∧
    ∀ n ∈ A, Set.HasPosDensity (UnrepWithout A n h) := by⊢ True ↔ ∃ A h, MinAsymptoticAddBasisOfOrder A h ∧ A.HasPosDensity ∧ ∀ n ∈ A, (UnrepWithout A n h).HasPosDensity
  sorryAll goals completed! 🐙


end Erdos330