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import FormalConjecturesUtil
open Filter
open scoped Pointwise
namespace Erdos868
The number of ways in which a natural n can be written as the sum of
o members of the set A.
noncomputable
def ncard_add_repr (A : Set ℕ) (o : ℕ) (n : ℕ) : ℕ :=
{ a : Fin o → ℕ | Set.range a ⊆ A ∧ ∑ i, a i = n }.ncardLet $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) \to \infty$ as $n \to \infty$, then must $A$ contain a minimal additive basis of order $2$?
Larsen and Larsen [LaLa26] answered this in the negative.
@[category research solved, AMS 5 11]
theorem erdos_868.parts.i :
answer(False) ↔ ∀ (A : Set ℕ), A.IsAsymptoticAddBasisOfOrder 2 →
atTop.Tendsto (fun n => ncard_add_repr A 2 n) atTop → ∃ B ⊆ A,
B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2 := ⊢ False ↔
∀ (A : Set ℕ),
A.IsAsymptoticAddBasisOfOrder 2 →
Tendsto (fun n => ncard_add_repr A 2 n) atTop atTop →
∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2
All goals completed! 🐙Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) > \epsilon \log n$ for large $n$ and an arbitrary fixed $\epsilon > 0$, then must $A$ contain a minimal additive basis of order $2$?
Larsen and Larsen [LaLa26] constructed a counterexample with $f(n) > c \log n$ for all large $n$.
@[category research solved, AMS 5 11]
theorem erdos_868.parts.ii :
answer(False) ↔ ∀ᵉ (A : Set ℕ) (ε > 0), A.IsAsymptoticAddBasisOfOrder 2 →
(∀ᶠ (n : ℕ) in atTop, ε * Real.log n < ncard_add_repr A 2 n) → ∃ B ⊆ A,
B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2 := ⊢ False ↔
∀ (A : Set ℕ),
∀ ε > 0,
A.IsAsymptoticAddBasisOfOrder 2 →
(∀ᶠ (n : ℕ) in atTop, ε * Real.log ↑n < ↑(ncard_add_repr A 2 n)) →
∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2
All goals completed! 🐙Erdős and Nathanson proved that this is true if $f(n) > (\log \frac{4}{3})^{-1} \log n$ for all large $n$.
@[category research solved, AMS 5 11]
theorem erdos_868.variants.fixed_ε :
answer(True) ↔ ∀ (A : Set ℕ), A.IsAsymptoticAddBasisOfOrder 2 →
(∀ᶠ (n : ℕ) in atTop, (Real.log (4 / 3))⁻¹ * Real.log n < ncard_add_repr A 2 n) → ∃ B ⊆ A,
B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2 := ⊢ True ↔
∀ (A : Set ℕ),
A.IsAsymptoticAddBasisOfOrder 2 →
(∀ᶠ (n : ℕ) in atTop, (Real.log (4 / 3))⁻¹ * Real.log ↑n < ↑(ncard_add_repr A 2 n)) →
∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2
All goals completed! 🐙Härtter and Nathanson proved that there exist additive bases which do not contain any minimal additive bases.
@[category research solved, AMS 5 11]
theorem erdos_868.variants.Hartter_Nathanson (o : ℕ) (ho : 1 < o) : ∃ (A : Set ℕ),
A.IsAsymptoticAddBasisOfOrder o ∧ ∀ B ⊆ A, B.IsAsymptoticAddBasisOfOrder o →
∃ b ∈ B, (B \ {b}).IsAsymptoticAddBasisOfOrder o := o:ℕho:1 < o⊢ ∃ A,
A.IsAsymptoticAddBasisOfOrder o ∧
∀ B ⊆ A, B.IsAsymptoticAddBasisOfOrder o → ∃ b ∈ B, (B \ {b}).IsAsymptoticAddBasisOfOrder o
All goals completed! 🐙
end Erdos868