FormalConjectures.ErdosProblems.«260»

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import FormalConjecturesUtil

Erdős Problem 260

Reference: erdosproblems.com/260


namespace Erdos260


open Filter

Let $a_1 < a_2 < \cdots$ be an increasing sequence such that $\frac{a_n}{n} \to \infty$. Is the sum $\sum_{n}^{\infty} \frac{a_n}{2^{a_n}}$ irrational?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_260 : answer(sorry) ↔
                  ∀ a : ℕ → ℤ, ∀ s : ℝ,
                  StrictMono a →
                  Tendsto (fun n => (a n : ℝ ) / n ) atTop atTop →
                  HasSum (fun n => (a n : ℝ ) / 2 ^ a n) s → Irrational s :=
  sorry


-- TODO: Add a proof of the theorem under the strong assumption $a_{n+1}-a_n → \infty$
-- TODO: Add a proof of the theorem under the strong assumption $a_n \gg n\sqrt{\log{n}\log{\log{n}}}$

end Erdos260