On the set of points represented by harmonic subseries
Vjekoslav Kovač
TL;DR
This work resolves a higher-dimensional version of Erdős–Graham’s convergence problem by proving that the set of points $\{ (\sum_{n\in A}\frac{1}{n},\sum_{n\in A}\frac{1}{n+1},\sum_{n\in A}\frac{1}{n+2}) : A\subset \mathbb N \text{ infinite}, \sum_{n\in A}\frac{1}{n}<\infty \}$ in $\mathbb R^3$ has a non-empty interior. The authors reduce the problem to a constructive topological game between Alice and Bob, reframe the sums via a linear change of variables to a perturbed vector series $\sum_n\big((1/n,2/n^2,2/n^3)+O(1/n^4)\big)$, and develop a sequence of preparatory games to build intuition and control. A key arithmetic lemma provides a matrix $M\in GL(3,\mathbb R)$ and finite index sets $S_j,T_j$ yielding precise leading-term asymptotics $c_j/n^j \mathbf e_j$, enabling a finite-sum decomposition that forces convergence to any target inside a constructed box. Combining these elements yields a rigorous, constructive strategy proving the interior exists and even identifies an explicit open ball inside the interior, with the result complemented by a concrete, computer-assisted calculation. This advances the understanding of achievement sets in higher dimensions and demonstrates a concrete, algorithmic pathway to realize interior structure in sums of perturbed 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 non-empty interior. This generalizes a two-dimensional result by Erdős and Straus.