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Statement

Theorem 1 (p. 1). "Let $r$ be a positive rational number. For every $x$ that is sufficiently large in terms of $r$, there is a set $\mathcal E$ of integers not exceeding $x$, such that $\sum_{n\in\mathcal E}1/n=r$ and

$$|\mathcal E|>(1-e^{-r})x-O_r\Bigl(\frac{x\log\log x}{\log x}\Bigr).$$

Furthermore, this is best possible: the main term cannot be increased, nor can the error term be reduced."

Here an Egyptian fraction is a sum of reciprocals of distinct positive integers (p. 1), so $\mathcal E$ is a set of distinct positive integers. The bound (display (2)) improves the author's earlier theorem that some such $\mathcal E$ has $|\mathcal E|>C(r)x$ for a positive constant $C(r)$ (the paper's [8]).

"Best possible" is made precise by Proposition 6 (p. 4): there is $\delta(r)>0$ such that for every $x$ sufficiently large in terms of $r$, every set $\mathcal E$ of positive integers not exceeding $x$ with $\sum_{n\in\mathcal E}1/n=r$ satisfies

$$|\mathcal E|\le(1-e^{-r})x-\delta(r)\,\frac{x\log\log x}{\log x}.$$

The paper notes (p. 2) that Croot's recent result, that $r$ has a representation with all denominators in $[x,e^rx+O_r(x\log\log x/\log x)]$ for large $x$ (the paper's [3]), implies Theorem 1.

Source. G. Martin, Denser Egyptian fractions, Acta Arith. 95 (2000), no. 3, 231--260 (DOI 10.4064/aa-95-3-231-260); read in the arXiv preprint arXiv:math/9811112v1 (18 November 1998), whose pagination is used here: Theorem 1 and display (2) on p. 1, Proposition 6 on p. 4, the deduction on pp. 4--5. The journal pagination was not compared.

Read depth. Claims checked: the statement and Proposition 6 were read clause by clause on the page images of pp. 1 and 4; the deduction of Theorem 1 from Propositions 5 and 6 (pp. 4--5) and the proof of Proposition 6 (Section 3, pp. 7--9) were read for structure. The proofs of Propositions 7 and 8 (Sections 4--5, pp. 10--19) were not read.

Proof pointer

Section 2 (pp. 4--5) reduces Theorem 1 to Propositions 5 and 6. For the lower bound, take $I=\{r\}$ and $t=\lceil(1-e^{-r})x-C(r)x\log\log x/\log x\rceil$ (display (8)) with $C(r)$ large; Proposition 5 gives a set of $t$ distinct positive integers with reciprocal sum $r$ and largest element below $t/(1-e^{-r})+O_r(t\log\log t/\log t)$, which is at most $x$ once $C(r)$ is large enough. The optimality of both terms is Proposition 6, proved in Section 3 (pp. 8--9) by showing through Lemma 9 that a denominator of such a representation has no prime factor much above $x/\log x$ (unless that prime divides the denominator of $r$) and counting the integers this excludes with Lemma 10.

Dependencies

Same-paper Propositions 5 and 6, and through them Propositions 7 and 8 and Lemmas 9--17; Proposition 5 combines Croot's techniques (the paper's [2]) with the author's earlier method (the paper's [8]).

Bears on

No Erdős problem page of this corpus is linked to this theorem. It is the fixed-bound counterpart of Theorem 2, which fixes the number of terms instead and bears on Problem 285.