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Claim. With $b(a)$ the least $b>a$ such that the lowest-terms denominator of $\sum_{a\le n\le b}1/n$ is smaller than that of $\sum_{a\le n\le b-1}1/n$ (the paper's convention, one more than the site's), and with

$$c=\sum_{d\ge1}\frac{\delta(f_d)}{d(d+1)},\qquad f_d(x)=\sum_{i=0}^d\prod_{\substack{j=0\\ j\ne i}}^d(x-j),$$

where $\delta(f_d)$ is the density of primes modulo which $f_d$ has a root, Theorem 1 of the preprint states

$$\liminf_{a\to\infty}\frac{b(a)-a}{\log a}=\frac1{1+c}\approx0.546.$$

The inequality $\ge$ is Lemma 31 of the author's 2024 paper; the preprint proves $\le$ in the stronger form that for every $C<1+c$ and all large $n$ there are integers $a,b>e^{Cn}$ with $b=a+n$ and a drop at $b$. With Lemma 32 of the 2024 paper ($0.82<c<0.85$) this gives infinitely many $a$ with $b(a)<a+0.55\log a$.

Submission note. Posted to erdosproblems.com as a proof claim by Wouter van Doorn (account Woett) on 2 September 2026, giving "GPT-5.6 Sol" as the AI used:

In [VD24] it was proven that $b(a) - a \gg \log a$, but that there are also infinitely many $a$ with $b(a) - a \ll \log a$. In particular, the lower limit

$$> \liminf_{a \to \infty} \frac{b(a) - a}{\log a} >$$

exists. Moreover, a specific lower bound (that turns out to be approximately $0.546$) was given. In this new paper we prove that this lower bound is tight, thereby pinning down the precise value of the lower limit. Notes: From the ideas in [VD24] it was clear that this lower limit result could be proven, if one could ensure that there exists an integer coming from a CRT solution with prime moduli, that also avoids certain residue classes modulo other primes. To give an example: if you have two residue classes mod $3$ and two residue classes mod $5$, then by CRT you obtain four integers in the interval $[1, 15]$. And then the question is: is at least one of these four integers even? ChatGPT came up with the idea of cleverly using an equality of Halász over $\mathbb{F}_p$ to guarantee something akin to this, and the rest of the proof uses arguments from [VD24]. This paper is a human-written simplification of the original ChatGPT write-up.

Covers. The exact value of the limit inferior of $(b(a)-a)/\log a$, the growth question at its smallest scale. It settles neither the existence question, which rests on the 2024 paper, nor any upper bound for $b(a)$, nor the typical size of $b(a)-a$.

Depends on. Van Doorn's 2024 paper supplies the lower bound $\ge1/(1+c)$ (its Lemma 31) and the constants.

Standing. Author preprint, arXiv:2609.00104v1 (31 August 2026, 9 pages), described on the card van Doorn 2026 with its Theorem 1 page; no journal record, citing paper or independent review was found on 2026-09-17, and the proof is compiled for structure only, so the claim is pending. The author filed it on the site's proof-claim tab on 2 September 2026 naming the system GPT-5.6 Sol; the paper's Section 4 credits ChatGPT 5.6-Sol Pro with finding the application of a Halász-type concentration inequality over $\mathbb F_p$ that secures the last residue condition of its Theorem 3, and describes the paper as a human-written simplification of the machine write-up, "The sharp lower-limit constant for the first decrease of a harmonic denominator", which the author's repository (created 29 August 2026) holds, linked above. The tab lists the entry without a full or partial label; the result concerns only the growth question, so the claim is partial, and its value is proved because the result is a proved identity, the exact value of the limit inferior, which determines the growth of $b(a)-a$ only at its smallest scale.