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Statement

Write $E_{m,k}(N)$ for the number of $n\le N$ for which $m/n$ cannot be written as a sum of $k$ unit fractions (p. 240). The survey recalls that Vaughan showed $E_{4,3}(N)\le N\exp(-c(\log N)^{2/3})$ for some positive $c$ (its [45]: R. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970)), and states the generalization:

Theorem 3 (Elsholtz [10]). For positive integers $m>k\ge3$ there is a constant $c_{m,k}>0$, depending only on $m$ and $k$, with

$$E_{m,k}(N)\ \le\ N\exp\Bigl(-c_{m,k}(\log N)^{1-\frac1{2^{k-1}-1}}\Bigr).$$

"Note that $m=4$ and $k=3$ recovers Vaughan's bound." The survey adds that Viola had a similar bound with $1/(k-1)$ in place of $1/(2^{k-1}-1)$, which Shen improved to $1/k$ (p. 240).

Source. Bloom and Elsholtz, Egyptian fractions, Nieuw Arch. Wiskd. (5) 23 (2022), no. 4, 237--245; Theorem 3 and the Vaughan sentence on p. 240 (PDF p. 4), read on the page image. The theorem is the survey's restatement of C. Elsholtz, Sums of $k$ unit fractions, Trans. Amer. Math. Soc. 353 (its reference [10]); neither Elsholtz's paper nor Vaughan's is held here, so both bounds are recorded second-hand from the survey.

Read depth. Claims checked: the statement and the Vaughan sentence were read clause by clause on the page image. The survey gives no proof; it describes the key idea (p. 240) as the realization that solutions of $m/n=1/x_1+\cdots+1/x_k$ can be parametrized so that sieve methods apply.

Dependencies

Elsholtz's paper (the survey's [10]) and, for the case $m=4$, $k=3$, Vaughan's paper (its [45]); neither held.

Bears on

• Problem 242: the site's Vaughan bound, "the number of exceptions in $[1,x]$ is $\le x\exp(-c(\log x)^{2/3})$", is the case $m=4$, $k=3$; a first-hand statement of the same bound with explicit dependence on $m$ is Pomerance and Weingartner's Theorem 1.3.