FormalConjectures.ErdosProblems.«347»

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import FormalConjecturesUtil

Erdős Problem 347

Reference: erdosproblems.com/347


open Filter Set Topology


namespace Erdos347

The set of subset sums of a set A ⊆ ℕ.

local notation "𝓟" A => subsetSums A

Is there a sequence $A={a_1\leq a_2\leq \cdots}$ of integers with $$\lim \frac{a_{n+1}}{a_n}=2$$ such that $$P(A')= \left{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right}$$ has density $1$ for every cofinite subsequence $A'$ of $A$?

This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and van Doorn, also in the comments).

Thos was formalized in Lean by Barschkis using Aristotle.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/ebarschkis/ErdosProblem/blob/main/Problem347/Formalization.lean"]
theorem declaration uses 'sorry'erdos_347 :
    answer(True) ↔ ∃ a : ℕ → ℕ, (Monotone a) ∧
      (Tendsto (fun n ↦ (a (n + 1) : ℝ) / (a n : ℝ)) atTop (𝓝 2)) ∧
      (∀ ι : ℕ → ℕ, (range ι)ᶜ.Finite → HasDensity (𝓟 (range (a ∘ ι))) 1) := by⊢ True ↔
  ∃ a,
    Monotone a ∧
      Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) atTop (𝓝 2) ∧
        ∀ (ι : ℕ → ℕ), (range ι)ᶜ.Finite → (𝓟 range (a ∘ ι)).HasDensity 1
  sorryAll goals completed! 🐙


end Erdos347