Problem Statement
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?
Does $X$ contain an open set?
Categories:
Number Theory
Progress
Erdős and Straus proved the answer is yes for the 2-dimensional version, where $X\subseteq \mathbb{R}^2$ is the set of\[\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1}\right) \]as $A\subseteq\mathbb{N}$ ranges over all infinite sets with $\sum_{n\in A}\frac{1}{n}<\infty$.The answer is yes, proved by Kovač [Ko24], who constructs an explicit open ball inside the set. Kovač and Tao [KoTa24] have proved an analogous result for all higher dimensions.
Source: erdosproblems.com/268 | Last verified: January 14, 2026