On the set of points represented by harmonic subseries

TL;DR

This paper provides a constructive solution to a problem in combinatorial number theory, demonstrating that the set of points formed by harmonic subseries over infinite sets has a non-empty interior, extending previous two-dimensional results.

Contribution

It introduces a new approach to show the set of harmonic subseries points has a non-empty interior, solving an open problem posed by Erdős and Graham.

Findings

The set of harmonic subseries points has a non-empty interior.

The result generalizes a two-dimensional case by Erdős and Straus.

A constructive winning strategy for the convergence game is provided.

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.