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import FormalConjecturesUtilErdős Problem 326
open Filter
open scoped Topology
namespace Erdos326
Let $A \subset \mathbb{N}$ be an additive basis of order 2.
Must there exist $B = {b_1 < b_2 < \dots} \subseteq A$ which is also a basis such that $\lim_{k\to\infty} \frac{b_k}{k^2}$ does not exist?
@[category research open, AMS 5 11]
theorem erdos_326 : answer(sorry) ↔ ∀ (A : Set ℕ), A.IsAddBasisOfOrder 2 →
∃ (b : ℕ → ℕ), StrictMono b ∧ ∀ n, b n ∈ A ∧ (Set.range b).IsAddBasis ∧
∀ (x : ℝ), ¬ Tendsto (fun n ↦ (b n : ℝ) / n ^ 2) atTop (𝓝 x) := ⊢ True ↔
∀ (A : Set ℕ),
A.IsAddBasisOfOrder 2 →
∃ b,
StrictMono b ∧
∀ (n : ℕ), b n ∈ A ∧ (Set.range b).IsAddBasis ∧ ∀ (x : ℝ), ¬Tendsto (fun n => ↑(b n) / ↑n ^ 2) atTop (𝓝 x)
All goals completed! 🐙
Erdős originally asked whether this was true with A = B, but this was disproved by Cassels.
-- Formalisation note: This is trivially true for `x = 0` by taking `a = id`. Cassels' proof
-- shows it for `0 < x` which is more interesting.
@[category research solved, AMS 5 11]
theorem erdos_326.variants.eq :
∃ (a : ℕ → ℕ) (_ : StrictMono a) (_ : Set.range a |>.IsAddBasisOfOrder 2) (x : ℝ) (_ : 0 < x),
Tendsto (fun n ↦ (a n : ℝ) / n ^ 2) atTop (𝓝 x) := ⊢ ∃ a,
∃ (_ : StrictMono a) (_ : (Set.range a).IsAddBasisOfOrder 2),
∃ x, ∃ (_ : 0 < x), Tendsto (fun n => ↑(a n) / ↑n ^ 2) atTop (𝓝 x)
All goals completed! 🐙
end Erdos326