Problem 268

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?

Source: erdosproblems.com/268

Number theory

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An accepted solution exists. The statement is true.

Proved. The site shows PROVED (LEAN); the parenthesis is a catalog label explained under Formalization. Kovač's Theorem 1 (arXiv:2405.07681, May 2024; Amer. Math. Monthly 132 (2025), 895--911, refereed) proves that $X$ has nonempty interior, the question's affirmative answer, and is recorded as the accepted claim Kovač 2024; Kovač and Tao's Corollary 2.10 (arXiv:2406.17593v3, November 2024; Acta Math. Hungar. 175 (2025), 572--608, refereed), the same statement in every dimension, is recorded as a second accepted claim, Kovač and Tao 2024. The two Lean developments that follow Kovač's proof are linked from his claim page and were not built here.

Problem 268

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