On the set of points represented by harmonic subseries

Abstract

We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erd\H{o}s and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after several reductions using peculiar arithmetic identities, the game outcome shows that the set of points \[ \Big(\sum_{n\in A}\frac{1}{n}, \sum_{n\in A}\frac{1}{n+1}, \sum_{n\in A}\frac{1}{n+2}\Big), \] obtained as $A$ ranges over infinite sets of positive integers, has a non-empty interior. This generalizes a two-dimensional result by Erd\H{o}s and Straus.