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Claim. Theorem 3 (p. 446) is printed as "For any $\varepsilon>0$, for any sufficiently large prime $p$ and any integer $c$ there exist $k=4\varepsilon^{-3}+O(\varepsilon^{-2})$ pairwise distinct integers $x_i$ with $1\le x_i\le p^\varepsilon$, $i=1,\ldots,k$, and such that the congruence (1) holds", where (1) is $\sum_{i=1}^k1/x_i\equiv c\pmod p$ (p. 445). So for $p\ge p_0(\varepsilon)$ every residue modulo $p$ is a sum of $k\ll\varepsilon^{-3}$ elements of $\{n^{-1}:1\le n\le p^\varepsilon\}$, even with pairwise distinct summands, a stronger form than Problem 1180 asks. For the finitely many primes $p<p_0(\varepsilon)$, every residue $a\in\{0,\ldots,p-1\}$ is the sum of $a$ copies of $1=1^{-1}$, at most $p-1<p_0(\varepsilon)$ summands, so $C_\varepsilon=\max(k,p_0(\varepsilon))$ answers the problem's question, which allows a summand to be repeated (the authored one-line remark of the problem page); the paper's abstract speaks of any prime $p$ while the theorem's wording keeps to large $p$, a looseness the theorem's wording corrects, since with distinct summands the primes with $p^\varepsilon<2$ cannot be covered. The theorem, quoted verbatim above, is compiled on the result page theorem_3; the digest is on the card shparlinski_2002_question_erdos_graham.

Argument, in outline. With $m=\lceil\varepsilon^{-1}+1/2\rceil$ and $k=2m^2(2m-1)+1$, the $x_i$ are found among the products of two primes from an interval $[X,2X]$ with $4X^2\le p^\varepsilon$: the number of solutions of (1) with such $x_i$ has main term $(\#\mathscr W(X))^k/p$ against an error controlled by Karatsuba's exponential-sum bound in the form of Friedlander and Iwaniec (the paper's Lemma 2) and the orthogonality of additive characters; repeated summands are removed and the count is positive for large $p$ by the prime number theorem. The outline covers the whole paper (pp. 445--448); Lemma 2 rests on Theorem 2 of Friedlander and Iwaniec, which is not held, so no step is checked against its inputs.

Acceptance. Refereed: Igor E. Shparlinski, On a question of Erdős and Graham, Arch. Math. (Basel) 78 (2002), no. 6, 445--448, received 30 August 2000; the publisher's record dates the issue 1 June 2002, the date this page is named by. Reviewed: the site's curator, Thomas F. Bloom, labels the problem proved and credits Shparlinski, in the problem's commentary, with answering the original question in the affirmative with $C_\epsilon\ll\epsilon^{-3}$; the curator neither wrote nor submitted the result. Croot's 2004 paper and Glibichuk's 2006 paper cite the theorem as the first answer. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the completion to the finitely many small primes is the one line above.