Let $X\subseteq {\mathbb{R}}^3$ be the set of all points of the shape$$\left(\sum _{n\in A}\frac{1}{n},\sum _{n\in A}\frac{1}{n+1},\sum _{n\in A}\frac{1}{n+2}\right)$$as $A\subseteq \mathbb{N}$ ranges over all infinite sets with ${\sum }_{n\in A}\frac{1}{n}<\infty$.Does $X$ contain an open set?

AMS 40 54 · solved (literature)

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