Glibichuk 2006 combinatorial properties sets residues modulo prime

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lemma_4: Glibichuk's 2006 lemma that for every real epsilon > 0, every integer m greater than 8([1/epsilon+1/2]+1)^2 and every sufficiently large prime p, the m-fold sumset of the inverses modulo p of the integers in [1, p^epsilon] is all of Z_p; summands may repeat, and Theorem 3 makes them distinct.

theorem_1: Glibichuk's 2006 sum-product theorem that for subsets A and B of the residues modulo a prime p, with B antisymmetric (B and -B disjoint) and |A||B| > p, every residue is a sum of eight products ab with a in A and b in B; Section 3 of the paper applies it to sums of inverses of primes to prove Lemma 4 and Theorem 3.

theorem_2: Glibichuk's 2006 sum-product theorem that for subsets A and B of the residues modulo a prime p, with B symmetric (B = -B) and |A||B| > p, every residue is a sum of eight products ab with a in A and b in B; the companion of Theorem 1 for symmetric sets.

theorem_3: Glibichuk's 2006 theorem that for every epsilon > 0, every sufficiently large prime p and every residue a modulo p there are pairwise distinct positive integers x_1, ..., x_N at most p^epsilon, N = 8([1/epsilon+1/2]+1)^2, whose inverses modulo p sum to a; the bound of order epsilon^{-2} for Problem 1180, improving Shparlinski's epsilon^{-3}.

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Glibichuk, A. A., Combinatorial properties of sets of residues modulo a prime and the {E}rdős-{G}raham problem. Mat. Zametki (2006), 384--395. The journal record is Mat. Zametki 79 (2006), no. 3, 384--395, DOI 10.4213/mzm2708; English translation Math. Notes 79 (2006), no. 3--4, 356--365, DOI 10.1007/s11006-006-0040-8 (both Crossref records read; the translation was not read). Received 3 May 2005, revised version 26 September 2005 (p. 395).

In Russian. The Erdos-Graham problem asks whether for every eps>0 there is k(eps) such that for all large primes p and every residue class c there are at most k(eps) distinct integers 1 <= x_i <= p^eps with sum of x_i^{-1} congruent to c mod p, where x^{-1} denotes the least positive inverse. The paper's main result, Theorem 3, gives N = 8([1/eps + 1/2] + 1)^2 pairwise distinct x_1,...,x_N <= p^eps with a congruent to x_1^{-1} + ... + x_N^{-1} mod p, improving Shparlinski's bound of 4eps^{-3} + O(eps^{-2}) obtained from Karatsuba's exponential-sum estimates (Croot had earlier given k <= log^{3+o(1)} p). The method is additive-combinatorial rather than analytic: Theorem 1 states that if A, B are subsets of Z_p with B antisymmetric (B intersect -B empty) and |A||B| > p, then 8AB = Z_p; Theorem 2 gives the same conclusion 8AB = Z_p when B is symmetric (B = -B) and |A||B| > p. The paper announces Theorem 3 as derived from these two sum-product statements together with the technique of Karatsuba's papers [4] and [6] (p. 385); its proof in Section 3 starts from Lemma 4 (p. 391: for m > 8([1/eps + 1/2] + 1)^2 and large p, every residue is a sum of m inverses of integers in [1, p^eps], repetition allowed), whose proof applies Theorem 1 (p. 393). This is the source for Problem 1180, the Erdos-Graham question on representing residues as short sums of inverses of small integers.

Source: https://www.mathnet.ru/eng/mzm2708.

The copy read and the introduction's attributions. The copy read for this card is the journal's twelve-page file (dvips and Ghostscript output) whose text layer is in a legacy Cyrillic encoding; printed p. $n$ is PDF p. $n-383$. The abstract, the introduction's first two pages with Theorems 1--3 (printed pp. 384--385; the introduction closes on p. 386) and the reference list (p. 395) were read on the page images (130 dpi renders) on 2026-09-18, with the formulas as the check. The introduction (p. 384) states the Erdős--Graham problem as the existence, for every $\varepsilon>0$, of $k(\varepsilon)$ such that for every sufficiently large prime $p$ and every integer $c$ some $k\le k(\varepsilon)$ pairwise distinct integers $1\le x_i\le p^\varepsilon$ satisfy $\sum_{i\le k}x_i^{-1}\equiv c\pmod p$ (1), and attributes to Croot [2] (On some questions of Erdős and Graham about Egyptian fractions, Mathematika 46 (1999), 359--372, filed as crootiii_1999_questions_erdos_graham_about_egyptian_fractions) the choice of $k\le\log^{3+o(1)}p$ pairwise distinct numbers in $[1,p^\varepsilon]$ satisfying (1); to Shparlinski [3] (printed "Sparlinski"; Arch. Math. 78 (2002), 445--448) the bound $k=4\varepsilon^{-3}+O(\varepsilon^{-2})$ for every $\varepsilon>0$ and sufficiently large $p$, from Karatsuba's trigonometric-sum estimates [4]--[6]; and to Croot [7] (Reciprocal power sums modulio [sic] a prime, e-print math.NT/0403360, 2004, the paper filed as croot_2004_sums_reciprocal_powers_modulo_prime) the existence, for every $\varepsilon\in(0,1]$ and every power $k\ge1$, of $N=N(k,\varepsilon)$ with positive integers $x_1,\ldots,x_N\le p^\varepsilon$ and $a\equiv(x_1^k)^{-1}+\cdots+(x_N^k)^{-1}\pmod p$ for every sufficiently large prime $p$ and every residue $a$ (Croot's own Theorem 2 states this for every prime $p\ge2$). Theorem 3 is announced as strengthening the result of [3]. The file prints "© А. А. Глибичук 2006" in the footer of its first page (printed p. 384, read on the page image), and the hosting site's Terms of Use (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02) state that "All materials published on this website including full-text articles, abstracts and author indexes are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that "Reproduction or republication of the materials contained on Math-Net.Ru in any form requires written permission of the copyright holder", naming no open license, every other right reserved.

Read status: claims checked for the abstract and Theorem 3 (printed pp. 384--385, PDF pp. 1--2) and for the introduction's attributions, read clause by clause on the page images; claims checked for Theorems 1 and 2 (p. 385, restated pp. 388--389), Definition 1 (p. 385) and Lemma 4 (p. 391), read clause by clause on the page images on 2026-10-08; the proofs (Sections 2--3, pp. 386--394) were read for structure only and not checked. Nothing here is independently reviewed.

Bears on. #1180 (Theorem 3, printed p. 385, PDF p. 2, page image: for every $\varepsilon>0$, every sufficiently large prime $p$ and every residue $a$, pairwise distinct positive integers $x_1,\ldots,x_N\le p^\varepsilon$ with $N=8([1/\varepsilon+1/2]+1)^2$ and $a\equiv x_1^{-1}+\cdots+x_N^{-1}\pmod p$, the site's $C_\varepsilon\ll\varepsilon^{-2}$ for $p$ large in terms of $\varepsilon$; Lemma 4, p. 391, page image: for each integer $m>8([1/\varepsilon+1/2]+1)^2$ and $p$ large in terms of $\varepsilon$, every residue is a sum of $m$ inverses of integers in $[1,p^\varepsilon]$, repetition allowed, as the problem's wording allows; the introduction's attributions to Croot and Shparlinski)

Results to transcribe.

• Theorem 1 (p. 385): If A, B are subsets of Z_p with B antisymmetric (B intersect (-B) empty) and |A||B| > p, then 8AB = Z_p. Paged as theorem_1.
• Theorem 2 (p. 385): If A, B are subsets of Z_p with B symmetric (B = -B) and |A||B| > p, then 8AB = Z_p. Paged as theorem_2.
• Lemma 4 (p. 391): For real eps > 0, A the set of inverses mod p of the integers in [1, p^eps], every integer m > 8([1/eps + 1/2] + 1)^2 and every sufficiently large prime p, mA = Z_p. Paged as lemma_4.
• Theorem 3 (p. 385): For every eps > 0, all sufficiently large primes p and every residue a mod p there exist distinct positive integers x_1,...,x_N <= p^eps with N = 8([1/eps + 1/2] + 1)^2 such that a is congruent to x_1^{-1} + ... + x_N^{-1} mod p. Paged as theorem_3.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.