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Mat. Zametki, 2006, Volume 79, Issue 3, Pages 384–395 (Mi mz2708)  

This article is cited in 32 scientific papers (total in 33 papers)

Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem

A. A. Glibichuk

M. V. Lomonosov Moscow State University

Abstract: Consider an arbitrary $\varepsilon>0$ and a sufficiently large prime $p>2$. It is proved that, for any integer $a$, there exist pairwise distinct integers $x_1,x_2,…,x_N$, where $N=8([1/\varepsilon+1/2]+1)^2$ such that $1\le x_i\le p^\varepsilon$, $i=1,…,N$, and
$$ a\equiv x_1^{-1}+\dotsb+x_N^{-1}\pmod p, $$
where $x_i^{-1}$ is the least positive integer satisfying $x_i^{-1}x_i\equiv1\pmod p$. This improves a result of Sparlinski.

DOI: https://doi.org/10.4213/mzm2708

Full text: PDF file (231 kB)
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English version:
Mathematical Notes, 2006, 79:3, 356–365

Bibliographic databases:

UDC: 511.3
Received: 03.05.2005
Revised: 26.09.2005

Citation: A. A. Glibichuk, “Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem”, Mat. Zametki, 79:3 (2006), 384–395; Math. Notes, 79:3 (2006), 356–365

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Garaev M. Z., “An explicit sum-product estimate in $\mathbb F_p$”, Int. Math. Res. Not. IMRN, 2007, rnm035, 11 pp.  crossref  mathscinet  zmath  isi  scopus
    2. Garaev M. Z., Garcia V. C., “Waring type congruences involving factorials modulo a prime”, Arch. Math. (Basel), 88:1 (2007), 35–41  crossref  mathscinet  zmath  isi  scopus
    3. Glibichuk A.A., Konyagin S.V., “Additive Properties of Product Sets in Fields of Prime Order”, Additive Combinatorics, CRM Proceedings & Lecture Notes, 43, eds. Grantville A., Nathanson M., Solymosi J., Amer Mathematical Soc, 2007, 279–286  crossref  mathscinet  isi
    4. M. Z. Garaev, V. C. Garcia, S. V. Konyagin, “Waring's problem with the Ramanujan $\tau$-function”, Izv. Math., 72:1 (2008), 35–46  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Garaev M. Z., Garcia V. C., “The equation $x_1x_2=x_3x_4+\lambda$ in fields of prime order and applications”, J. Number Theory, 128:9 (2008), 2520–2537  crossref  mathscinet  zmath  isi  elib  scopus
    6. Ayyad A., Cochrane T., “Lattices in $\mathbb Z^2$ and the congruence $xy+UV\equiv c\pmod m$”, Acta Arith., 132:2 (2008), 127–133  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Hart D., Losevich A., “Sums and Products in Finite Fields: an Integral Geometric Viewpoint”, Radon Transforms, Geometry, and Wavelets, Contemporary Mathematics, 464, eds. Olafsson G., Grinberg E., Larson D., Jorgensen P., Massopust P., Quinto E., Rubin B., Amer Mathematical Soc, 2008, 129–135  crossref  mathscinet  zmath  isi
    8. Glibichuk A., Rudnev M., “On additive properties of product sets in an arbitrary finite field”, J. Anal. Math., 108 (2009), 159–170  crossref  mathscinet  zmath  isi  elib  scopus
    9. Covert D., Hart D., Iosevich A., Koh D., Rudnev M., “Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields”, Eur. J. Comb., 31:1 (2010), 306-319  crossref  mathscinet  zmath  isi  elib  scopus
    10. I. D. Shkredov, “Fourier analysis in combinatorial number theory”, Russian Math. Surveys, 65:3 (2010), 513–567  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. M. Z. Garaev, “Sums and products of sets and estimates of rational trigonometric sums in fields of prime order”, Russian Math. Surveys, 65:4 (2010), 599–658  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    13. Hart D., Iosevich A., Koh D., Rudnev M., “Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture”, Trans. Amer. Math. Soc., 363:6 (2011), 3255–3275  crossref  mathscinet  zmath  isi  elib  scopus
    14. Garcia V.C., Luca F., Mejia Huguet V.J., “On Sums of Fibonacci Numbers Modulo P”, Bull. Aust. Math. Soc., 83:3 (2011), 413–419  crossref  mathscinet  zmath  isi  elib  scopus
    15. I. V. Vyugin, I. D. Shkredov, “On additive shifts of multiplicative subgroups”, Sb. Math., 203:6 (2012), 844–863  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. S. A. Gritsenko, E. A. Karatsuba, M. A. Korolev, I. S. Rezvyakova, D. I. Tolev, M. E. Changa, “Scientific Achievements of Anatolii Alekseevich Karatsuba”, Proc. Steklov Inst. Math., 280, suppl. 2 (2013), S1–S22  mathnet  crossref  crossref  zmath  isi  elib
    17. Proc. Steklov Inst. Math., 280, suppl. 2 (2013), S23–S29  mathnet  crossref  crossref  zmath  isi  elib
    18. Schoen T., Shkredov I.D., “Additive Properties of Multiplicative Subgroups of F-P”, Q. J. Math., 63:3 (2012), 713–722  crossref  mathscinet  zmath  isi  elib  scopus
    19. Chapman J., Erdogan M.B., Hart D., Iosevich A., Koh D., “Pinned Distance Sets, K-Simplices, Wolff's Exponent in Finite Fields and Sum-Product Estimates”, Math. Z., 271:1-2 (2012), 63–93  crossref  mathscinet  zmath  isi  elib  scopus
    20. Covert D., Iosevich A., Pakianathan J., “Geometric Configurations in the Ring of Integers Modulo P(l)”, Indiana Univ. Math. J., 61:5 (2012), 1949–1969  crossref  mathscinet  zmath  isi  scopus
    21. Alnaser Ala' Jamil, “Bounds for Waring's Number Mod P(M) in Number Fields”, J. Number Theory, 133:1 (2013), 72–82  crossref  mathscinet  zmath  isi  scopus
    22. Garcia V.C., “On the Distribution of Sparse Sequences in Prime Fields and Applications”, J. Theor. Nr. Bordx., 25:2 (2013), 317–329  crossref  mathscinet  zmath  isi  scopus
    23. Le Anh Vinh, “On Four-Variable Expanders in Finite Fields”, SIAM Discret. Math., 27:4 (2013), 2038–2048  crossref  mathscinet  zmath  isi  scopus
    24. Shkredov I.D., “On Exponential Sums Over Multiplicative Subgroups of Medium Size”, Finite Fields their Appl., 30 (2014), 72–87  crossref  mathscinet  zmath  isi  scopus
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    26. Cochrane T., Hart D., Pinner Ch., Spencer C., “Waring's Number For Large Subgroups of $\mathbb Z^*_p$”, Acta Arith., 163:4 (2014), 309–325  crossref  mathscinet  zmath  isi  scopus
    27. Shparlinski I.E., “Points on Varieties Over Finite Fields in Small Boxes”, Scholar - a Scientific Celebration Highlighting Open Lines of Arithmetic Research, Contemporary Mathematics, 655, ed. Cojocaru A. David C. Pappalardi F., Amer Mathematical Soc, 2015, 209–233  crossref  mathscinet  zmath  isi
    28. Diaz C.A., Garaev M.Z., “Sums of fractions modulo p”, Arch. Math., 106:4 (2016), 337–344  crossref  mathscinet  zmath  isi  scopus
    29. Ayyad A., “On the Congruence Ax - By Equivalent to C (Mod P) and the Finite Field Z(P)”, Notes Number Theory Discret. Math., 22:1 (2016), 29–32  zmath  isi
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    31. Macourt S., Shkredov I.D., Shparlinski I.E., “Multiplicative Energy of Shifted Subgroups and Bounds on Exponential Sums With Trinomials in Finite Fields”, Can. J. Math.-J. Can. Math., 70:6 (2018), 1319–1338  crossref  mathscinet  isi  scopus
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