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Tr. Mosk. Mat. Obs., 2013, Volume 74, Issue 1, Pages 35–73 (Mi mmo540)  

This article is cited in 19 scientific papers (total in 19 papers)

Some new results on higher energies

I. D. Shkredovab

a Steklov Mathematical Institute of the Russian Academy of Sciences
b Delone Laboratory of Discrete and Computational Geometry, P. G. Demidov Yaroslavl State University

Abstract: This article is concerned with the method of higher energies from combinatorial number theory. Upper bounds are obtained for the additive energies of convex sets and of sets $A$ with small $|AA|$ and $|A(A+1)|$. New structural results, involving the notion of a dual popular difference set, are proved in terms of higher energies.

Key words and phrases: combinatorial number theory; higher energies; popular difference set.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 11.G34.31.0053
Russian Foundation for Basic Research 11-01-00759_а

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English version:
Transactions of the Moscow Mathematical Society, 2013, 74, 31–63

Bibliographic databases:

UDC: 511.37
MSC: 11H99
Received: 30.12.2012
Revised: 17.02.2013

Citation: I. D. Shkredov, “Some new results on higher energies”, Tr. Mosk. Mat. Obs., 74, no. 1, MCCME, M., 2013, 35–73; Trans. Moscow Math. Soc., 74 (2013), 31–63

Citation in format AMSBIB
\by I.~D.~Shkredov
\paper Some new results on higher energies
\serial Tr. Mosk. Mat. Obs.
\yr 2013
\vol 74
\issue 1
\pages 35--73
\publ MCCME
\publaddr M.
\jour Trans. Moscow Math. Soc.
\yr 2013
\vol 74
\pages 31--63

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    This publication is cited in the following articles:
    1. Roche-Newton O., Warren A., “New Expander Bounds From Affine Group Energy”, Discret. Comput. Geom.  crossref  isi  scopus
    2. Shkredov I.D., “on Exponential Sums Over Multiplicative Subgroups of Medium Size”, Finite Fields their Appl., 30 (2014), 72–87  crossref  mathscinet  zmath  isi  elib  scopus
    3. Ilya Sh., “Energies and Structure of Additive Sets”, Electron. J. Comb., 21:3 (2014), P3.44  zmath  isi
    4. I. D. Shkredov, “Structure theorems in additive combinatorics”, Russian Math. Surveys, 70:1 (2015), 113–163  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. I. D. Shkredov, “On sums of Szemerédi–Trotter sets”, Proc. Steklov Inst. Math., 289 (2015), 300–309  mathnet  crossref  crossref  isi  elib
    6. I. D. Shkredov, “On a question of A. Balog”, Pac. J. Math., 280:1 (2016), 227–240  crossref  mathscinet  zmath  isi  elib  scopus
    7. T. Schoen, I. D. Shkredov, “Additive dimension and a theorem of Sanders”, J. Aust. Math. Soc., 100:1 (2016), 124–144  crossref  mathscinet  zmath  isi  scopus
    8. I. V. Vyugin, E. V. Solodkova, I. D. Shkredov, “On the Additive Energy of the Heilbronn Subgroup”, Math. Notes, 101:1 (2017), 58–70  mathnet  crossref  crossref  mathscinet  isi  elib
    9. B. Murphy, O. Roche-Newton, I. D. Shkredov, “Variations on the Sum-Product Problems II”, SIAM Discret. Math., 31:3 (2017), 1878–1894  crossref  mathscinet  zmath  isi  scopus
    10. I. D. Shkredov, “On asymptotic formulae in some sum-product questions”, Trans. Moscow Math. Soc., 2018, 231–281  mathnet  crossref  mathscinet  elib
    11. I. D. Shkredov, “Differences of subgroups in subgroups”, Int. J. Number Theory, 14:4 (2018), 1111–1134  crossref  mathscinet  zmath  isi  scopus
    12. I. D. Shkredov, D. Zhelezov, “On additive bases of sets with small product set”, Int. Math. Res. Notices, 2018, no. 5, 1585–1599  crossref  mathscinet  zmath  isi  scopus
    13. 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  scopus
    14. K. I. Olmezov, “Sharpening an Estimate of the Size of the Sumset of a Convex Set”, Math. Notes, 107:6 (2020), 954–957  mathnet  crossref  crossref  mathscinet  isi  elib
    15. 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
    16. K. I. Olmezov, “Additive Properties of Slowly Increasing Convex Sets”, Math. Notes, 108:6 (2020), 827–841  mathnet  crossref  crossref  mathscinet  isi  elib
    17. Rudnev M., Shakan G., Shkredov I.D., “Stronger Sum-Product Inequalities For Small Sets”, Proc. Amer. Math. Soc., 148:4 (2020), 1467–1479  crossref  mathscinet  zmath  isi  scopus
    18. K. I. Olmezov, “An Elementary Analog of the Operator Method in Additive Combinatorics”, Math. Notes, 109:1 (2021), 110–119  mathnet  crossref  crossref  mathscinet  isi
    19. I. D. Shkredov, “Nekommutativnye metody v additivnoi kombinatorike i teorii chisel”, UMN, 76:6(462) (2021), 119–180  mathnet  crossref
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