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Claim. Theorem 3 (p. 385; in Russian): for every $\varepsilon>0$, every sufficiently large prime $p$ and every residue class $a\pmod p$ there are positive pairwise distinct integers $x_1,\ldots,x_N\le p^\varepsilon$ with $N=8([1/\varepsilon+1/2]+1)^2$ and $a\equiv x_1^{-1}+\cdots+x_N^{-1}\pmod p$. So $C_\varepsilon\le8([1/\varepsilon+1/2]+1)^2\ll\varepsilon^{-2}$ for $p\ge p_0(\varepsilon)$, even with 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 $C_\varepsilon=\max(8([1/\varepsilon+1/2]+1)^2,p_0(\varepsilon))$ answers the problem's question, which allows a summand to be repeated (the authored one-line remark of the problem page). The theorem is compiled on the result page theorem_3; the digest is on the card glibichuk_2006_combinatorial_properties_sets_residues_modulo_prime.

Argument, as stated. The proof (Sections 2--3, pp. 386--394) rests on the paper's Theorems 1 and 2, sum-product statements of the form $8AB=\mathbb Z_p$ for $|A||B|>p$ with $B$ antisymmetric or symmetric, proved with the technique of Bourgain, Katz and Tao; the paper announces (p. 385) that Theorem 3 combines them with Karatsuba's technique, but the proof (pp. 391--394) uses only Theorem 1, through Lemma 4, together with Karatsuba's technique and Chebyshev's lower bound for the number of primes. The proof is not checked here; the result page gives a pointer to it. The introduction (p. 384) credits the first answer to Shparlinski, on Shparlinski's page, and the earlier bound of $\log^{3+o(1)}p$ distinct summands to Croot's 1999 paper.

Acceptance. Refereed: A. A. Glibichuk, Combinatorial properties of sets of residues modulo a prime and the Erdős--Graham problem, Mat. Zametki 79 (2006), no. 3, 384--395, received 3 May 2005 and revised 26 September 2005; English translation Math. Notes 79 (2006), no. 3--4, 356--365, not held. The Russian record gives the year only and the translation's record dates its issue to March 2006, so the page is named by the first of that month. Reviewed: the site's curator, Thomas F. Bloom, labels the problem proved and credits Glibichuk, in the problem's commentary, with the improvement to $C_\epsilon\ll\epsilon^{-2}$; the curator neither wrote nor submitted the result. The theorem is cited in the sum-product literature (eighteen citing records on Semantic Scholar). Nothing here is independently reviewed by this project.

Formalization. A Lean file in Boris Alexeev's lean-proofs repository, linked above at a pinned commit, declares Glibichuk as its informal author and Codex and GPT-5.6 Sol as its formal authors, says its proof follows this paper, and states erdos_1180: for every $\varepsilon>0$ there is a $C$ such that every residue modulo every prime $p$ is the sum of a list of at most $C$ inverses of admissible integers in $[1,p^\varepsilon]$, so the small primes are included. The statement file for the problem in formal-conjectures names that declaration as its formal proof. The file contains no sorry and has not been built or audited here, so no formalized evidence is listed.

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