Problem 268

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claims/: The 2 claim pages of Problem 268, one per claimant's result; the problem's standing derives from them.

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Statement. Let $X\subseteq \mathbb{R}^3$ be the set of all points of the shape

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

as $A\subseteq\mathbb{N}$ ranges over all infinite sets with $\sum_{n\in A}\frac{1}{n}<\infty$. Does $X$ contain an open set?

Status. Proved. The site shows PROVED (LEAN); the parenthesis is a catalog label explained under Formalization. Kovač's Theorem 1 (arXiv:2405.07681, May 2024; Amer. Math. Monthly 132 (2025), 895--911, refereed) proves that $X$ has nonempty interior, the question's affirmative answer, and is recorded as the accepted claim Kovač 2024; Kovač and Tao's Corollary 2.10 (arXiv:2406.17593v3, November 2024; Acta Math. Hungar. 175 (2025), 572--608, refereed), the same statement in every dimension, is recorded as a second accepted claim, Kovač and Tao 2024. The two Lean developments that follow Kovač's proof are linked from his claim page and were not built here.

Source. erdosproblems.com/268, accessed 2026-09-04 (the problem page: PROVED (LEAN); last edited 28 September 2025; source keys [ErGr80, p. 65] and [Er88c, p. 105]; commentary citing [Ko24] and [KoTa24]) and 2026-10-07 (its two-comment discussion thread of 13 April and 26 May 2026 and its empty proof-claim tab). Cite as: T. F. Bloom, Erdős Problem #268, https://www.erdosproblems.com/268, accessed 2026-10-07.

References.

• [Ko24] Kova\v{c}, V., On the set of points represented by harmonic subseries. arXiv:2405.07681v3 (2024); Amer. Math. Monthly 132 (2025), 895--911, DOI 10.1080/00029890.2025.2540753.
• [KoTa24] Kova\v{c}, V. and Tao, T., On several irrationality problems for Ahmes series. arXiv:2406.17593v4; Acta Math. Hungar. 175 (2025), 572--608, DOI 10.1007/s10474-025-01528-0.

Formalization. The site's (LEAN) suffix is a catalog label. The formal-conjectures declaration erdos_268 (at the linked commit of 18 September 2026, the last to touch the file by 2026-10-07) states the theorem for every dimension $d$, that the set of $d$-tuples $(\sum_{n\in A}1/(n+i))_{i<d}$ over infinite $A$ with convergent reciprocal sum has nonempty interior, cites Kovač and Tao for it, and has a sorry body. Its formal_proof attribute pins a gist at an exact revision whose declaration covers only the three-dimensional set, whose header names Matteo Del Vecchio and Aristotle (Harmonic) as authors and says that it follows Kovač's paper, and whose source uses native_decide in one arithmetic step. The thread's first comment (13 April 2026) reports this autoformalization from Kovač's paper, and its second (26 May 2026) links the repository Jayyhk/erdos-lean, where the same development appears without native_decide. Both files are linked from Kovač's claim page as formalizations of his proof, at the linked revisions. Neither artifact was built or audited line by line here, so these are formal-source records rather than formal-proof credit.

Current assessment

Kovač's Theorem 1 in arXiv:2405.07681v3 proves exactly that the set $X$ in the question has non-empty interior. The statement recorded here is that of v3; the later American Mathematical Monthly record establishes publication, but its Version of Record was not compared line by line with the preprint.

The proof is not compiled here, and neither formal artifact was built or audited line by line. The claim pages record the two refereed proofs; the curator credits Kovač with the solution and cites Kovač and Tao for the analogous result in every dimension.

Search scope (2026-10-07 UTC). The site's problem page, discussion thread and proof-claim tab; the formal-conjectures declaration at the commit linked under Formalization and the two Lean developments at the revisions linked from Kovač's claim page; the arXiv records of 2405.07681 and 2406.17593 and the Crossref records of both publications. None found a dispute of the theorem. Not searched: MathSciNet, zbMATH, Google Scholar, X.

Progress

The method overview on p. 2 applies a linear change of variables, obtaining a perturbed vector series with leading coordinates $(1/n,2/n^2,2/n^3)+O(1/n^4)$, and introduces a convergence game. Section 3 and Lemma 2 begin on p. 8, the proof of Theorem 1 begins in Section 4 on p. 10 and ends on p. 13, and Section 5 computes an explicit ball of radius $10^{-24}$ on p. 14. These locators describe the argument's scope; the proof is not compiled here.

Known Results

• Kovač, Theorem 1. For infinite $A\subseteq\mathbb N$ with $\sum_{n\in A}1/n<\infty$, the set of triples $(\sum_{n\in A}1/n,\sum_{n\in A}1/(n+1),\sum_{n\in A}1/(n+2))$ has non-empty interior in $\mathbb R^3$. This is the exact assertion of Problem 268.
• Kovač--Tao, Theorem 2.8 and Corollaries 2.9--2.10. For every positive integer $d$, Theorem 2.8 gives a $\beta>1$ such that the $d$-tuples of shifted reciprocal sums from strictly increasing sequences with $\lim_{k\to\infty}a_k^{1/\beta^k}=\infty$ form a set with non-empty interior. Corollary 2.9 obtains one such sequence for which all $d$ shifted sums are rational. Corollary 2.10 then proves the unrestricted $d$-dimensional interior theorem over all infinite $A$ with convergent reciprocal sum. Its $d=3$ case is a later extension of the exact result already proved directly by Kovač.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• kovac_2024_several_irrationality_problems_ahmes_series
• kovac_2024_set_points_represented_harmonic_subseries
• kovac_2024_set_points_represented_harmonic_subseries / lemma_2
• kovac_2024_set_points_represented_harmonic_subseries / theorem_1