Kovac 2024 set points represented harmonic subseries

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lemma_2: Kovač's arithmetic lemma: there are an invertible 3 by 3 matrix M, six mutually disjoint finite sets of positive integers and positive constants c_1, c_2, c_3 such that signed sums of M applied to (1/(an), 1/(an+1), 1/(an+2)) equal c_j/n^j times the j-th basis vector up to O(1/n^4).

theorem_1: Kovač's theorem that the set of triples of the sums of 1/n, 1/(n+1) and 1/(n+2) over infinite sets A of positive integers with convergent sum of 1/n has non-empty interior in R^3; Section 5 of the paper reports an explicit ball of radius 10^-24 inside the set.

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Vjekoslav Kovač, On the set of points represented by harmonic subseries. arXiv:2405.07681v3, 12 September 2024. Published in The American Mathematical Monthly 132 (2025), 895--911, DOI 10.1080/00029890.2025.2540753.

The inspected theorem artifact is arXiv v3. The publication metadata was checked separately; the full Version of Record was not compared line by line.

Erdős and Graham (their 1980 monograph, p. 65) report an unpublished theorem of Erdős and Straus, that the pairs $(\sum_k1/a_k,\sum_k1/(1+a_k))$ over sequences of integers with $\sum_k1/a_k<\infty$ contain an open set, the sequences being understood, as the paper reads the context, as positive and strictly increasing; they ask whether the same holds in three or more dimensions. Theorem 1 (p. 1) answers the three-dimensional question positively: the set of triples

$$\left(\sum_{n\in A}\frac1n,\sum_{n\in A}\frac1{n+1}, \sum_{n\in A}\frac1{n+2}\right)$$

over infinite $A\subseteq\mathbb N$ (the positive integers) with $\sum_{n\in A}1/n<\infty$ has non-empty interior in $\mathbb R^3$.

The p. 2 overview makes a linear change of variables and obtains a perturbed vector series with leading coordinates $(1/n,2/n^2,2/n^3)+O(1/n^4)$, and recasts the problem as an infinite convergence game against an adversary who chooses the perturbation errors. Section 2 (pp. 3--8) warms up with simpler games, including a sketch of the two-dimensional Erdős--Straus case. Section 3 (pp. 8--9) proves Lemma 2, an arithmetic lemma with an explicit matrix $M$, six disjoint finite sets and constants $c_1,c_2,c_3$. Section 4 (pp. 10--13) proves Theorem 1: a cautious greedy strategy reaches every point of a rectangular box, and $M^{-1}$ maps that box to a non-degenerate parallelepiped inside the set. Section 5 (pp. 13--14) runs the construction numerically and reports, on p. 14, a ball of radius $10^{-24}$ inside the set.

Source: https://arxiv.org/abs/2405.07681. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2405.07681), every other right reserved.

Read status. Claims checked: Theorem 1 (p. 1), Lemma 2 with its explicit data (pp. 8--9) and the Section 5 report (pp. 13--14) were read clause by clause on the printed pages of arXiv v3, and the power-sum identities behind Lemma 2's constants were recomputed. The proof of Theorem 1 (pp. 10--13) was read but not checked step by step, and the Section 5 numerics were not recomputed.

Bears on. #268: the set in Theorem 1 is the set $X$ of the problem as worded, so Theorem 1 as stated in arXiv v3 is the affirmative answer to that question; Lemma 2 bears on it only as a step of that proof.

Results. Theorem 1 (p. 1), with the explicit ball of Section 5 (p. 14) recorded on its page; Lemma 2 (p. 8). The games of Section 2 and Claims 1 and 2 of Section 4 (pp. 11--12) are motivation and proof steps, summarized on the Theorem 1 page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.