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Kovač and Tao prove, as Theorem 2.8, that for every positive integer $d$ there is $\beta>1$ such that the $d$-tuples $(\sum_k1/a_k,\sum_k1/(a_k+1),\ldots,\sum_k1/(a_k+d-1))$ over strictly increasing sequences with $a_k^{1/\beta^k}\to\infty$ fill a set with nonempty interior, and draw from it Corollary 2.10: for every $d$ the set of such $d$-tuples over all infinite $A\subseteq\mathbb N$ with $\sum_{n\in A}1/n<\infty$ has nonempty interior. At $d=3$ this is the set $X$ of Problem 268, so the corollary answers the question yes a second time, by a different route, through the rapidly growing sequences of Theorem 2.8, from the explicit ball of Kovač's page. Corollary 2.9 adds, for every $d$, one such sequence for which all $d$ shifted sums are rational.

The paper's digest is the library card Kovač and Tao 2024.

Read depth. The statements are those of the arXiv v4 text; the proofs are not compiled here, so this project has made no independent review of them.

Acceptance. The paper is refereed: V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), 572--608. The site's commentary cites the paper for the analogous result in every higher dimension and credits the solution of the problem itself to Kovač's earlier paper, so no curator review of this result as a solution is listed. The page is dated by the arXiv posting of 27 November 2024, the third version, which first carries these results and first names Tao as coauthor.