On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with
$\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n \sim F(n)$.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
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theorem erdos_264.variants.ko_tao_pos {F : ℕ → ℕ}
(hF : atTop.Tendsto (fun n ↦ (F (n + 1) : ℝ) / F n) atTop) :
∃ a : ℕ → ℕ, IsIrrationalitySequence a ∧ (fun n ↦ (a n : ℝ)) ~[atTop] fun n ↦ (F n : ℝ)
