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Statement

Notation (p. 2 and p. 1): $w(A)=\sum_{1<a\in A}1/a$; $\mathrm{Div}(N)$ is the set of divisors of $N$; $A\cdot B=\{ab:a\in A,\ b\in B\}$; $\ll$ is the standard asymptotic notation the paper uses throughout.

Hypothesis 3 (p. 5), with parameters $\mathcal D\subset\mathbb N$, $y\in\mathbb N$ and $0<\beta\le1$: for every integer $h\in[y/2,\,y^{\lceil10/\beta\rceil-2}]$,

$$\sum_{d\in\mathcal D}h_d^2/d^2>y^{1/4},$$

where $h_d$ is the distance from $h$ to $d\mathbb Z$. The paper describes it as the condition needed for the intermediate frequency range of the circle-method argument.

Theorem 4 (p. 5), restated. Let $\ell/k\in(0,1]$, $\epsilon>0$ and $\beta>0$ be constants, and let $\mathcal L$ be an integer sufficiently large in terms of $\epsilon$ and $\beta$. Let $y_1,\ldots,y_s$ be positive integers greater than $\mathcal L$ such that for every $i<s$, $y_i\le y_{i+1}/2$ and

$$\bigl[y_i^{\lceil4/\beta\rceil-5/4},\,e^{y_i/\log^2y_i}\bigr]\cap \bigl[y_{i+1}^{\lceil4/\beta\rceil-5/4},\,e^{y_{i+1}/\log^2y_{i+1}}\bigr]\ne\emptyset.$$

Let each $\mathcal Q_i$ be a set of integers between $y_i$ and $2y_i$, such that the elements of $\mathcal Q:=\bigcup_i\mathcal Q_i$ are pairwise coprime. Let $\mathcal B$ be a set of integers dividing $(y_1^2)!$ that contains $1$ and whose elements are coprime to every element of $\mathcal Q$. Let $E\subset\mathcal B$ contain at least every element of $\mathcal B$ less than $y_1^2$, and put

$$\mathcal D=\Bigl(\mathcal B\cdot\mathrm{Div}\Bigl(\prod_{q\in\mathcal Q}q\Bigr)\Bigr)\setminus E.$$

Let $m$ be the least common multiple of $\mathcal D$, assume $k\mid m$, and put $\alpha=w(\mathcal D)(\ell/k)^{-1}$. If

$$1+\frac{\epsilon}{100}\le\alpha\le\log^{1+\epsilon}y_1, \qquad \frac{y_i^\beta}{\log^2y_i}\ll\lvert\mathcal Q_i\rvert\le y_i^\beta \ \text{ for all } i,$$

and Hypothesis 3 holds for $y=y_1$, then there exist at least two subsets $\mathcal D'$ of $\mathcal D$ with $w(\mathcal D')\equiv\ell/k\pmod 1$.

Hypothesis 3 is stated for $0<\beta\le1$ while Theorem 4 assumes only $\beta>0$; the paper applies Theorem 4 with $\beta=1$ (p. 7) and $\beta=1/2$ (p. 9), and in both it checks Hypothesis 3 for the set $\mathcal D$ of the application.

Source. D. Larsen, Sufficiently abundant numbers are pseudoperfect, 9-page manuscript (GitHub Larsen-Daniel/Erdos-318, 318.pdf, commit 39139e2b of 1 February 2026); Hypothesis 3 and Theorem 4 on p. 5, proof of Theorem 4 on pp. 5--7.

Read depth. Claims checked: Hypothesis 3 and Theorem 4 were read clause by clause on the page image of p. 5. The proof was read for structure only (below) and is not verified here.

Proof pointer and sketch (pp. 5--7)

The proof counts subsets $\mathcal D'$ with the weight $(\alpha-1)^{\lvert\mathcal D\rvert-\lvert\mathcal D'\rvert}$ and expands the congruence condition in additive characters modulo $m$, so the weighted count is a sum over frequencies $h$ of a product $f(h)$ over $d\in\mathcal D$. The frequency $h=0$ gives the main term. Large frequencies are shown to contribute an exponentially small amount unless $h$ lies near a multiple of a product of the $\mathcal Q_i$; small frequencies keep the real part of $f(h)$ positive; and the intermediate range is exactly where Hypothesis 3 is used. The weighted count, divided by $\max(1,\alpha-1)^{\lvert\mathcal D\rvert}$, bounds the number of solutions from below, and that bound exceeds $1$, which gives at least two subsets.

Dependencies

No other numbered result of the paper; the proof on pp. 5--7 uses the orthogonality of additive characters modulo $m$ and elementary estimates.

Standing

An unrefereed manuscript with a declared AI-assistance acknowledgment (proofreading, p. 9), read statically; no journal record was found on 2026-09-18. Consumers state the theorem with this qualification.

Bears on. #318: Theorem 4 is the tool the paper applies in its proof of Theorem 6, the squares case of the problem; it does not state that case itself. #825: Theorem 4 is the final step of the proof of Theorem 1, from which Corollary 5 gives the problem's statement.