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Statement

For a positive rational $r$ and a positive integer $j$, let

$$\mathcal L_j(r)=\{x\in\mathbb Z,\ x>r^{-1}:\text{ there are no integers } x_1>\cdots>x_t\ge1\text{ with }\textstyle\sum_{i=1}^t1/x_i=r\text{ and }x_j=x\},$$

so that $\mathcal L_j(r)$ holds the integers above $r^{-1}$ that never stand in position $j$ when the denominators of a representation of $r$ are listed in decreasing order (p. 3). An integer $x\le r^{-1}$ has $1/x\ge r$, so it can occur only in the one-term representation $r=1/x$; the paper leaves such $x$ out.

Theorem 3 (p. 3). "Let $r$ be a positive rational number. The set $\mathcal L_j(r)$ is finite for any integer $j\ge2$, and there exists an integer $j_0(r)$ such that $\mathcal L_j(r)$ is empty for all $j\ge j_0(r)$."

The paper's own consequences (p. 3): $\mathcal L_2(1)$ is finite, and "possibly $\{2,4\}$ is a complete list of integers (greater than $1$) with this property"; $\mathcal L_j(1)$ is empty once $j$ is large, and "possibly this holds for every $j\ge3$". The paper does not compute $\mathcal L_2(1)$ or $\mathcal L_3(1)$.

Source. G. Martin, Denser Egyptian fractions, arXiv:math/9811112v1 (18 November 1998), Theorem 3 on p. 3, read on the page image and in the text layer of that preprint; the proof is Section 6 (pp. 20--24). The journal version, Acta Arith. 95 (2000), no. 3, 231--260, was not compared.

Read depth. Claims checked: the statement and the definition of $\mathcal L_j(r)$ were read clause by clause on the page image of p. 3. Of the proof, pp. 20--21 (Lemmas 18 and 19, the construction for $\mathcal L_2(r)$ and the passage to $j\ge3$) were read for structure on the page images; the proof of Lemma 19, through Lemma 20 (pp. 21--24), was not checked.

Proof pointer

Section 6 (p. 20) restates Proposition 5 as Lemma 18: for a closed interval $I\subset(0,\infty)$ there is $X(I)$ such that for all $x>X(I)$ and all $r=a/b\in I$ with $P^*(b)<x\log^{-23}x$ there is a set $E$ of positive integers not exceeding $x$ with $\sum_{n\in E}1/n=r$. Lemma 19 (deduced on p. 22 from Lemma 20, which is stated on p. 21 and proved on pp. 22--24) gives, for every integer $k>k_0$, a positive integer $K\equiv-1\pmod k$ with $P^*(K)<k\log^{-24}k$. The proof of Theorem 3 begins (p. 20) by showing that $\mathcal L_2(r)$ is finite: for $r=a/b$, $I=[r/2,r]$ and a large integer $k$, take $K$ from Lemma 19. The remainder $r'=r-1/k-1/(Kk)$ lies in $I$, and since $1/k+1/(Kk)=((K+1)/k)/K$ with $(K+1)/k$ an integer, its denominator has no prime-power divisor above $k\log^{-24}k$; so Lemma 18 with $x=k/\log k$ represents $r'$ by integers at most $x$, and adding $k$ and $Kk$ gives a representation of $r$ with largest denominator $Kk$ and second-largest denominator $k$ (pp. 20--21). The higher positions use a different argument (p. 21): splitting the term with the largest denominator by identity (4) gives $\mathcal L_2(r)\supset\mathcal L_3(r)\supset\cdots$, and for each $n\in\mathcal L_2(r)$ a representation of $r-1/n$ by denominators exceeding $n$ shows $n\notin\mathcal L_j(r)$ for some $j$, so the sets are eventually empty.

Dependencies

Same-paper Proposition 5 (through Lemma 18) and Lemmas 19--20.

Bears on

• Problem 292: the site's "Explore $A$" invites finer questions. The analogue for the second-largest and later denominators is not posed in the 1980 monograph; the paper says it is mentioned in Guy's Unsolved problems in number theory (p. 3). Theorem 3 answers it for representations of $1$ up to finitely many exceptions, while the density question itself is Theorem 4.