---

Source. Theorem 2 (Refined version), Section 1.3, p. 2 of arXiv v3 (11 June 2024); the proof in Section 3, pp. 8--12 (Section 3.3, p. 12, applies its Lemma 7), by the techniques the paper credits on p. 2 to Danicic, Harman, Heilbronn and Hooley. Read on the PDF pages. The journal version, Mathematika 71 (2025), no. 2, e70009, was not compared.

Statement

Every $\varepsilon>0$ admits infinitely many pairs $(m,n)$ of positive integers with

$$\Bigl|\sum_{\ell=n}^{m}\frac1\ell-1\Bigr|\ \le\ \frac{1}{n^2(\log n)^{(5/4)-\varepsilon}}.$$

The statement is printed with the absolute value (p. 2, read on the rendered page image). The paper adds that one could further enforce $\sum_{\ell=n}^m1/\ell>1$ but chose not to, for simplicity; the remark is stated without proof, and the bound transfers to the overshoot $\varepsilon_n$ of Problem 314 only for pairs whose sum is at least $1$. The proof is non-constructive, resting on the existence of rational approximations of a special form. Section 1.3 contrasts the bound with a random model in which $X_n$ is uniform on $[0,1/n]$, under which only finitely many $n$ would have $X_n\le1/(n^2(\log n)^{1+\delta})$ (the model with $\varepsilon_n$ uniform on $[0,1/(en)]$ is Section 1.1's heuristic, p. 1): the theorem shows the random heuristic fails at this scale.

Proof pointer and sketch

The reduction of Part 1 and Part 2 of the proof of Theorem 1 turns the problem into approximating a real number by rationals of the form $a/b^2$; the trivial bound gives Theorem 1. Lemma 4 (p. 8) sharpens the estimate to $r_{3k+2}^{-1}=2k+3+O(1/k)$; the Erdős--Turán inequality (Lemma 5, p. 9) yields a count of $n$ with $pn^2/q$ close to a target modulo $1$ in a residue class (Lemma 6, pp. 9--11), hence infinitely many approximations $|\alpha-m/n^2|<n^{-5/2+\varepsilon}$ with prescribed residues of $m$ and $n$ for irrational $\alpha>0$ (Lemma 7, p. 11); applied to $\alpha=3/\sinh(1)$ in Lemma 8 (p. 12), this gives the logarithmic saving. These steps were read for structure only; no rewritten proof and no independent review exist here.

Dependencies and read depth

Same-paper: the reduction of Theorem 1's proof. External: the Erdős--Turán inequality in the form of Montgomery's book (Lemma 5, p. 9), with Baker's book cited for its consequence Lemma 6 (p. 9), the divisor bound (p. 10) and the transcendence of $e$ (p. 12). Heilbronn (1948), Danicic (1958), Hooley (1990) and Harman (1996) are cited on p. 2 for the techniques; Section 3 invokes none of their theorems. Read depth: claims checked; proof not verified.

Bears on. #314 (within the paper's framing; the bound is on $|\sum_{\ell=n}^m1/\ell-1|$ and reaches the overshoot $\varepsilon_n$ only for pairs whose sum is at least $1$, which the paper says could be enforced but does not prove); #288 (adjacent context; no exact integer sums).