FormalConjectures.ErdosProblems.«143»

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import FormalConjecturesUtil

Erdős Problem 143

Reference: erdosproblems.com/143

open Filter Finsetopen scoped Topologynamespace Erdos143

Let $A \subseteq (1, \infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k \geq 1$ we have $|kx - y| \geq 1$.

def WellSeparatedSet (A : Set ℝ) : Prop :=
  (A ⊆ (Set.Ioi (1 : ℝ))) ∧ Set.Infinite A ∧ Set.Countable A ∧
  (∀ x ∈ A, ∀ y ∈ A, x ≠ y → (∀ k ≥ (1 : ℕ), 1 ≤ |k * x - y|))

Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$

@[category research open, AMS 11]
theorem erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A →
    liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0 := by⊢ True ↔ ∀ (A : Set ℝ), WellSeparatedSet A → liminf (fun x ↦ ↑(A ∩ Set.Icc 1 x).ncard / x) atTop = 0
  sorryAll goals completed! 🐙

Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$

@[category research open, AMS 11]
theorem erdos_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) :
    Summable fun (x : A) ↦ 1 / (x * Real.log x) := byA:Set ℝh:WellSeparatedSet A⊢ Summable fun x ↦ 1 / (↑x * Real.log ↑x)
  sorryAll goals completed! 🐙

-- TODO(firsching): add the two other conjectures.
/-
$$
\sum_{\substack{x < n \\ x \in A}} \frac{1}{x} = o(\log n)?
$$

Perhaps even

$$
\sum_{\substack{x < n \\ x \in A}} \frac{1}{x} \ll \frac{\log x}{\sqrt{\log \log x}}?
$$
-/

end Erdos143