Lim 2024 differences two harmonic numbers

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theorem_1: States that for every c > 0 infinitely many pairs (m, n) have the sum of 1/l over n <= l <= m between 1 and 1 + c/n^2, so liminf n^2 eps(n) = 0 in the Erdős–Graham question listed as Problem 314.

theorem_2: States that for every epsilon > 0 infinitely many pairs (m, n) have the sum of 1/l over n <= l <= m within 1/(n^2 (log n)^(5/4 - epsilon)) of 1 in absolute value, by approximation of reals by rationals of the form a/b^2.

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Jeck Lim, Stefan Steinerberger, On differences of two harmonic numbers. arXiv preprint (2024). arXiv:2405.11354.

The copy read for this card is the thirteen-page arXiv version v3 (11 June 2024; v1 18 May 2024, v2 30 May 2024). The paper has since appeared as Mathematika 71 (2025), no. 2, e70009, doi:10.1112/mtk.70009, published 27 January 2025; the journal version was not obtained or compared, and the locators below are the preprint's. The preprint's Theorem 2 (p. 2, read on the rendered page image) is stated for the absolute value $|\sum_{\ell=n}^m1/\ell-1|$, as the site's commentary for Problem 314 and a comment of 22 January 2026 in its discussion thread report for the published version; the paper adds, without proof, that the sum could further be forced above $1$. Read status: Theorems 1 and 2 (pp. 1--2) and the quoted Erdős--Graham passage (p. 1) were read clause by clause on the PDF pages (claims checked); the proofs (Sections 2--3) were read for structure only and none has been independently reviewed. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2405.11354), every other right reserved.

The paper answers a question of Erdos and Graham on how small the excess eps_n = sum_{k=n}^{t} 1/k - 1 can be, where t is least with the sum at least 1, proving that liminf n^2 eps_n = 0. Theorem 1 gives an elementary construction: for every c > 0 there are infinitely many pairs (m,n) with 1 <= sum_{l=n}^{m} 1/l <= 1 + c/n^2, built from a rescaled subsequence of the continued fraction convergents of e together with harmonic-number asymptotics. Theorem 2 refines this non-constructively to |sum_{l=n}^{m} 1/l - 1| <= 1/(n^2 (log n)^{5/4 - eps}) for infinitely many pairs, using approximation of reals by rationals of the form a/b^2 in the style of Heilbronn, Danicic, Harman and Hooley. The paper itself frames the question as Erdos problem 314. For problem 288 the results are adjacent context: they concern the reciprocal sum over one interval coming close to 1, and give no exact integer sum and nothing about two intervals.

Source: https://arxiv.org/abs/2405.11354.

Bears on. #314 (the paper's own framing: on p. 1 it says Theorem 1 addresses the question's first half, $\liminf_n n^2\varepsilon_n=0$, and suffices to resolve it; on p. 2 it leaves open whether $n^{2+\delta}\varepsilon_n\to\infty$; Theorem 2 bounds $|\sum_{\ell=n}^m1/\ell-1|$ and reaches $\varepsilon_n$ only for pairs whose sum is at least $1$), #288 (adjacent context: one interval with reciprocal sum near $1$; no exact integer sum and no second interval).

Results.

• Theorem 1: Every c > 0 admits infinitely many pairs (m,n) of positive integers with 1 <= sum_{l=n}^{m} 1/l <= 1 + c/n^2, via convergents of the continued fraction of e.
• Theorem 2: Every eps > 0 admits infinitely many pairs (m,n) with |sum_{l=n}^{m} 1/l - 1| <= 1/(n^2 (log n)^{5/4 - eps}), using techniques for approximation by rationals a/b^2; the paper remarks, without proof, that the sum could also be forced above 1.

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