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Mat. Sb., 2012, Volume 203, Number 6, Pages 81–100 (Mi msb7857)  

This article is cited in 30 scientific papers (total in 30 papers)

On additive shifts of multiplicative subgroups

I. V. Vyugina, I. D. Shkredovbc*

a Institute for Information Transmission Problems, Russian Academy of Sciences
b Steklov Mathematical Institute, Russian Academy of Sciences
c Laboratory of Discrete and Computational Geometry named after B. N. Delone of P. G. Demidov Yaroslavl State University

Abstract: It is proved that for an arbitrary subgroup $R\subseteq\mathbb Z/p\mathbb Z$ and any distinct nonzero elements $\mu_1,…,\mu_k$ we have
$$ |R\cap(R+\mu_1)\cap…\cap(R+\mu_k)| \ll_k|R|^{{1}/{2}+\alpha_k} $$
under the condition that $1\ll_k|R|\ll_kp^{1-\beta_k}$, where $\{\alpha_k\}$$\{\beta_k\}$ are some sequences of positive numbers such that $\alpha_k,\beta_k\to0$ as $k\to\infty$. Furthermore, it is shown that the inequality $|R\pm R|\gg|R|^{5/3}\log^{-1/2}|R|$ holds for any subgroup $R$ such that $|R|\ll p^{1/2}$.
Bibliography: 25 titles.

Keywords: multiplicative subgroups, Stepanov's method, additive combinatorics.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00384
Ministry of Education and Science of the Russian Federation НШ-8508.2010.1

* Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2012, 203:6, 844–863

Bibliographic databases:

UDC: 511.218+511.336
MSC: Primary 11B75; Secondary 05B10, 11B13, 11T24
Received: 22.02.2011

Citation: I. V. Vyugin, I. D. Shkredov, “On additive shifts of multiplicative subgroups”, Mat. Sb., 203:6 (2012), 81–100; Sb. Math., 203:6 (2012), 844–863

Citation in format AMSBIB
\by I.~V.~Vyugin, I.~D.~Shkredov
\paper On additive shifts of multiplicative subgroups
\jour Mat. Sb.
\yr 2012
\vol 203
\issue 6
\pages 81--100
\jour Sb. Math.
\yr 2012
\vol 203
\issue 6
\pages 844--863

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    This publication is cited in the following articles:
    1. Bourgain J., Garaev M.Z., Konyagin S.V., Shparlinski I.E., “On the hidden shifted power problem”, SIAM J. Comput., 41:6 (2012), 1524–1557  crossref  mathscinet  zmath  isi  elib  scopus
    2. Shparlinski I.E., “Additive decompositions of subgroups of finite fields”, SIAM J. Discrete Math., 27:4 (2013), 1870–1879  crossref  mathscinet  zmath  isi  elib  scopus
    3. I. D. Shkredov, “Some new results on higher energies”, Trans. Moscow Math. Soc., 74 (2013), 31–63  mathnet  crossref  mathscinet  zmath  elib
    4. Schoen T., Shkredov I.D., “Higher moments of convolutions”, J. Number Theory, 133:5 (2013), 1693–1737  crossref  mathscinet  zmath  isi  elib  scopus
    5. Konyagin S.V., Rudnev M., “On new sum-product-type estimates”, SIAM J. Discrete Math., 27:2 (2013), 973–990  crossref  mathscinet  zmath  isi  elib  scopus
    6. D. Hart, “A note on sumsets of subgroups in $\mathbb Z_p^*$”, Acta Arith., 161:4 (2013), 387–295  crossref  mathscinet  isi  elib  scopus
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    8. I. Shkredov, “Energies and structure of additive sets”, Electron. J. Comb., 21:3 (2014), P3.44  mathscinet  zmath  isi
    9. I. D. Shkredov, “Sumsets in quadratic residues”, Acta Arith., 164:3 (2014), 221–243  crossref  mathscinet  zmath  isi  elib  scopus
    10. T. Cochrane, D. Hart, Ch. Pinner, C. Spencer, “Waring's number for large subgroups of $\mathbb Z_{p^*}^*$”, Acta Arith., 163:4 (2014), 309–325  crossref  mathscinet  zmath  isi  elib  scopus
    11. B. Murphy, O. Roche-Newton, I. Shkredov, “Variations on the Sum-Product Problem”, SIAM J. Discrete Math., 29:1 (2015), 514–540  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. A. Uvakin, “On Two-Dimensional Sums and Differences”, Math. Notes, 98:4 (2015), 636–652  mathnet  crossref  crossref  mathscinet  isi  elib
    13. T. Schoen, I. D. Shkredov, “Additive dimension and a theorem of Sanders”, J. Aust. Math. Soc., 100:1 (2016), 124–144  mathnet  crossref  mathscinet  zmath  isi
    14. I. V. Vyugin, E. V. Solodkova, I. D. Shkredov, “Intersections of Shifts of Multiplicative Subgroups”, Math. Notes, 100:2 (2016), 189–198  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    15. O. Roche-Newton, M. Rudnev, I. D. Shkredov, “New sum-product type estimates over finite fields”, Adv. Math., 293 (2016), 589–605  mathnet  crossref  mathscinet  zmath  isi  scopus
    16. O. E. Raz, M. Sharir, I. D. Shkredov, “On the number of unit-area triangles spanned by convex grids in the plane”, Comput. Geom., 62 (2017), 25–33  mathnet  crossref  mathscinet  zmath  isi  scopus
    17. M.-C. Chang, “Arithmetic progressions in multiplicative groups of finite fields”, Israel J. Math., 222:2 (2017), 631–643  crossref  mathscinet  zmath  isi  scopus
    18. E. A. Yazici, B. Murphy, M. Rudnev, I. Shkredov, “Growth estimates in positive characteristic via collisions”, Int. Math. Res. Not. IMRN, 2017, no. 23, 7148–7189  crossref  mathscinet  zmath  isi
    19. A. A. Uvakin, “On Two-Dimensional Sums in Abelian Groups”, Math. Notes, 103:2 (2018), 271–289  mathnet  crossref  crossref  mathscinet  isi  elib
    20. Rudnev M., “On the Number of Incidences Between Points and Planes in Three Dimensions”, Combinatorica, 38:1 (2018), 219–254  crossref  mathscinet  zmath  isi  scopus
    21. Shkredov I.D., “Differences of Subgroups in Subgroups”, Int. J. Number Theory, 14:4 (2018), 1111–1134  crossref  mathscinet  zmath  isi  scopus
    22. Shkredov I.D., Zhelezov D., “On Additive Bases of Sets With Small Product Set”, Int. Math. Res. Notices, 2018, no. 5, 1585–1599  crossref  mathscinet  zmath  isi
    23. Karpinski M., Merai L., Shparlinski I.E., “Identity Testing and Interpolation From High Powers of Polynomials of Large Degree Over Finite Fields”, J. Complex., 49 (2018), 74–84  crossref  mathscinet  zmath  isi  scopus
    24. 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  zmath  isi  scopus
    25. Murphy B. Petridis G. Roche-Newton O. Rudnev M. Shkredov I.D., “New Results on Sum-Product Type Growth Over Fields”, Mathematika, 65:3 (2019), 588–642  crossref  mathscinet  zmath  isi
    26. I. V. Vyugin, “A Bound for the Number of Preimages of a Polynomial Mapping”, Math. Notes, 106:2 (2019), 203–211  mathnet  crossref  crossref  mathscinet  isi  elib
    27. Murphy B. Rudnev M. Shkredov I.D. Shteinikov Yu.N., “On the Few Products, Many Sums Problem”, J. Theor. Nr. Bordx., 31:3 (2019), 573–602  mathscinet  zmath  isi
    28. Shkredov I.D., “Any Small Multiplicative Subgroup Is Not a Sumset”, Finite Fields their Appl., 63 (2020), UNSP 101645  crossref  mathscinet  isi
    29. K. I. Olmezov, A. S. Semchankau, I. D. Shkredov, “On Popular Sums and Differences for Sets with Small Multiplicative Doubling”, Math. Notes, 108:4 (2020), 557–565  mathnet  crossref  crossref  mathscinet  isi  elib
    30. Konyagin S.V., Makarychev S.V., Shparlinski I.E., Vyugin I.V., “On the Structure of Graphs of Markoff Triples”, Q. J. Math., 71:2 (2020), 637–648  crossref  mathscinet  zmath  isi
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