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Statement

Theorem 1 (p. 2). "For any $m,n\in\mathbb N$ and any $\epsilon>0$ there are at most $\mathcal O_\epsilon\bigl(n^\epsilon(\frac{n^3}{m^2})^{1/5}\bigr)$ solutions of the equation

$$\frac mn=\frac1{a_1}+\frac1{a_2}+\frac1{a_3}$$

in positive integers $a_1$, $a_2$ and $a_3$."

The paper notes (p. 2) that this improves Browning and Elsholtz's bound $O_\epsilon(n^\epsilon(n/m)^{2/3})$ in the range $m\ll n^{1/4}$. Its counting convention for $k$ unit fractions is nondecreasing tuples $a_1\le\cdots\le a_k$ (the function $f_k(m,n)$, p. 2); for a bound of this shape the ordering convention changes only the implied constant.

Source. Elsholtz and Planitzer, arXiv:1805.02945v1 (8 May 2018), 21 pp.; Theorem 1 on p. 2, read on the page image; proved on pp. 9--10, in Section 5 (pp. 8--11), with the patterns and relative greatest common divisors of Section 4 (pp. 6--8). Published as Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427, DOI 10.1017/prm.2018.137, online 30 January 2019 (Crossref record fetched); the published version was not compared.

Read depth. Claims checked: Theorem 1, Corollaries 1 and 2, Theorems 2 and 3 and Corollary 4 and Theorem 4 (pp. 2--4) were read clause by clause on the page images of pp. 2 and 4 and in the text layer of p. 3; the proofs were not read. The proof of Theorem 1 (pp. 9--10) was later read for its structure only and was not checked step by step.

Proof pointer

The proof (pp. 9--10, after the set-up of Section 5 on pp. 8--9) parametrizes the solutions with a fixed pattern $(n_1,n_2,n_3)$, $n_i=(a_i,n)$, through the relative greatest common divisors of Section 4. Inequality (18) (p. 10) bounds a product of five factors drawn from four quantities, $y$, $z$, $x_{12}x_{13}$ and $x_{12}x_{123}$ (the last squared), by $\ll n^3/m^2$, so one of the four is $O\bigl((n^3/m^2)^{1/5}\bigr)$; in each case the divisor bound $d(n)\ll_\epsilon n^\epsilon$ (Lemma A, p. 9) leaves $O_\epsilon(n^\epsilon)$ choices for the rest, and there are $O_\epsilon(n^\epsilon)$ patterns. Section 3 (pp. 4--5) gives the heuristic, attributed to Heath-Brown, that $f_3(m,n)=O_\epsilon(n^\epsilon)$ should hold.

Dependencies

The divisor bound (Lemma A, p. 9) and the parametrization of Section 5 through the relative greatest common divisors of Section 4; not examined here.

Bears on

• Problem 242: through Corollary 1 (the case $m=4$), an upper bound $O_\epsilon(n^{3/5+\epsilon})$ for the number of solutions for every $n$; an upper bound says nothing about existence.