erdos_347

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).

theorem 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)

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