erdos_247

Let $n_1 < n_2 < \cdots$ be a sequence of integers such that $$ \limsup \frac{n_k}{k} = \infty. $$ Is $$ \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} $$ transcendental?

theorem erdos_247 : answer(sorry) ↔ ∀ (n : ℕ → ℕ), (StrictMono n) →
    atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ →
    Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k)

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