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 Erdős 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 nonempty interior. This generalizes a twodimensional result by Erdős and Straus.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.07681
 arXiv:
 arXiv:2405.07681
 Bibcode:
 2024arXiv240507681K
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics
 EPrint:
 v2: 14 pages