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Claim. Croot's Main Theorem (On unit fractions with denominators in short intervals, Acta Arith. 99 (2001), no. 2, 99--114; arXiv:math/9904181, 30 April 1999): for every rational $r>0$ and every $N>1$ there are integers $N<x_1<\cdots<x_k\le(e^r+O_r(\log\log N/\log N))N$ with $r=1/x_1+\cdots+1/x_k$, and the error term is best possible. Its introduction poses the width question of Problem 286 in the corrected form $\min\{x_k-x_1\}\sim k$ and says that the theorem answers it for infinitely many $k$.

Covers. The site's question, an interval of width $(e-1+o(1))k$ containing the $k$ denominators of a representation of $1$, for the infinitely many $k$ that occur as the number of terms of a representation with denominators in $(N,(e+o(1))N]$, $N>1$: such a representation has $k>N$ terms, since each is below $1/N$, so its width, below $(e-1+o(1))N$, is below $(e-1+o(1))k$. These $k$ are infinitely many because $k>N$. The paper itself claims the sharper width $(1+o(1))k$ for infinitely many $k$, which rests on the construction using all but a vanishing proportion of the integers of the interval and has not been checked beyond the statement. Not covered: the question for every $k\ge2$, which neither the theorem nor the introduction asserts; the answer for every large $k$ follows from Martin's Theorem 2 and is the full claim Martin 1998.

Acceptance. The paper is published in Acta Arithmetica, a refereed journal (Crossref record of DOI 10.4064/aa99-2-1), which is the refereed evidence. The site's curator, Thomas Bloom, who is independent of the author, marks the problem PROVED and credits Croot's theorem as the proof in the commentary, which is the reviewed evidence; the commentary carries no qualification of the quantifier over $k$, and the scope recorded above follows the paper's own. The deduction under Covers. is written on this page and on the problem page from the theorem's statement; it is not in the paper. Proof coverage: the proof's structure is recorded from the preprint; it has not been verified, and the published proof has not been compared with the preprint's.

Related. The same theorem is the source of the accepted claim Croot 1999 on Problem 284, the first question of Croot's pair, where an elementary splitting step passes to every $k$. The public Lean proof of this problem's statement for every large $k$, in Boris Alexeev's repository, derives it from Martin's bound and is linked from the Martin page.