Erdős Problem 268

Reference:

• erdosproblems.com/268
• [KoTa24] Kova\vC, V. and Tao T., On several irrationality problems for Ahmes series.

theorem Erdos268.erdos_268 (d : ℕ) : (interior {x : Fin d → ℝ | ∃ (A : Set ℕ), A.Infinite ∧ (Summable fun (n : ↑A) => 1 / ↑↑n) ∧ x = fun (i : Fin d) => ∑' (n : ↑A), 1 / (↑↑n + ↑↑i)}).Nonempty

Let X be the set of points in Fin d → ℝ of the shape fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i) for some infinite subset A ⊆ ℕ such that 1 / n is summable over A. X has nonempty interior. This is proved in [KoTa24].