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Statement

$f_k(m,n)$ is the number of $k$-tuples $(a_1,\ldots,a_k)\in\mathbb N^k$ with $a_1\le a_2\le\cdots\le a_k$ and $m/n=1/a_1+\cdots+1/a_k$, for fixed $m,n\in\mathbb N$ (p. 2).

Theorem 2 (p. 2). For every $\epsilon>0$,

$$f_4(m,n)\ \ll_\epsilon\ n^\epsilon\Bigl(\frac{n^{4/3}}{m^{2/3}} +\frac{n^{28/17}}{m^{8/5}}\Bigr),$$

and for every $k\ge5$

$$f_k(m,n)\ \ll_\epsilon\ (kn)^\epsilon \Bigl(\frac{k^{4/3}n^2}{m}\Bigr)^{\frac{28}{17}\cdot2^{k-5}}.$$

The paper prints the statement without an explicit quantifier on $\epsilon$; the implied constants depend on $\epsilon$ only, as the subscript shows.

The paper compares this (p. 3) with Browning and Elsholtz's bounds $f_4(m,n)\ll_\epsilon n^\epsilon(n^{4/3}/m^{2/3}+(n/m)^{5/3})$ and, for $k\ge5$, the same shape as above with exponent $\frac53\cdot2^{k-5}$ in place of $\frac{28}{17}\cdot2^{k-5}$, noting $28/17=1.64705\ldots$.

Source. Christian Elsholtz and Stefan Planitzer, The number of solutions of the Erdős-Straus equation and sums of $k$ unit fractions, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427, read in arXiv:1805.02945v1 (8 May 2018), as identified on the source card; Theorem 2 on p. 2, proved on pp. 14--16 in Section 6 (pp. 11--16). The published version was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 2, and the comparison on p. 3. The proof was read for its structure only and was not checked step by step.

Proof pointer

The proof starts from the recursion (21) (p. 12), which bounds $f_k(m,n)$ by a sum of $f_{k-1}$ values over the possible smallest denominators. For $k=4$ it splits that sum at $u=n^\delta$: the short part is bounded with Browning and Elsholtz's $f_3(m,n)\ll_\epsilon n^\epsilon(n/m)^{2/3}$ (Lemma B, p. 12), and the long part, where the smallest denominator is large and forces the next one to be small, is bounded by $n^\epsilon n^{(12-4\delta)/5}/m^{8/5}$ through the four-variable pattern parametrization and the divisor bound (pp. 14--15). The bound $f_5(m,n)\ll_\epsilon n^\epsilon(n^2/m)^{28/17}$ follows from (21) (display (35), p. 16), and Lemma C (p. 12), a lifting procedure going back to Browning and Elsholtz, carries a bound $f_5(m,n)\ll_\epsilon n^\epsilon(n^2/m)^c$ with $c>1$ to every $k\ge5$; the paper takes $c=28/17$ (p. 16).

Dependencies

Lemma B (Browning and Elsholtz's three-term bound, p. 12), Lemma C (p. 12), the divisor bound (Lemma A, p. 9) and the pattern parametrization of Section 4 (pp. 6--8); not examined here.

Bears on

• Problem 148: through Corollary 3, which takes $m=n=1$; the problem's $F(k)$ counts only distinct denominators, so $F(k)\le f_k(1,1)$. The bound says nothing about lower bounds.