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Trudy Mat. Inst. Steklova, 2015, Volume 290, Pages 304–316 (Mi tm3634)  

This article is cited in 33 scientific papers (total in 34 papers)

On sum sets of sets having small product set

S. V. Konyagin, I. D. Shkredov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We improve the sum–product result of Solymosi in $\mathbb R$; namely, we prove that $\max \{|A+A|,|AA|\}\gg |A|^{4/3+c}$, where $c>0$ is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets $A\subset \mathbb R$ with $|AA| \le |A|^{4/3}$.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 290:1, 288–299

Bibliographic databases:

ArXiv: 1503.05771
UDC: 511.178
Received: March 15, 2015

Citation: S. V. Konyagin, I. D. Shkredov, “On sum sets of sets having small product set”, Modern problems of mathematics, mechanics, and mathematical physics, Collected papers, Trudy Mat. Inst. Steklova, 290, MAIK Nauka/Interperiodica, Moscow, 2015, 304–316; Proc. Steklov Inst. Math., 290:1 (2015), 288–299

Citation in format AMSBIB
\by S.~V.~Konyagin, I.~D.~Shkredov
\paper On sum sets of sets having small product set
\inbook Modern problems of mathematics, mechanics, and mathematical physics
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2015
\vol 290
\pages 304--316
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 290
\issue 1
\pages 288--299

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    This publication is cited in the following articles:
    1. S. V. Konyagin, I. D. Shkredov, “New results on sums and products in $\mathbb R$”, Proc. Steklov Inst. Math., 294 (2016), 78–88  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. B. Lund, “An improved bound on $(A+A)/(A+A)$”, Electron. J. Comb., 23:3 (2016), P3.46  mathscinet  zmath  isi  elib
    3. O. Roche-Newton, M. Rudnev, I. D. Shkredov, “New sum-product type estimates over finite fields”, Adv. Math., 293 (2016), 589–605  crossref  mathscinet  zmath  isi  elib  scopus
    4. O. Roche-Newton, “If $(A+A)/(A+A)$ is small, then the ratio set is large”, J. Lond. Math. Soc.-Second Ser., 93:1 (2016), 83–100  crossref  mathscinet  zmath  isi  scopus
    5. O. E. Raz, M. Sharir, I. D. Shkredov, “On the number of unit-area triangles spanned by convex grids in the plane”, Comput. Geom.-Theory Appl., 62 (2017), 25–33  crossref  mathscinet  zmath  isi  scopus
    6. I. D. Shkredov, “Some Remarks on the Balog–Wooley Decomposition Theorem and Quantities $D^{+}$, $D^\times$”, Proc. Steklov Inst. Math., 298, suppl. 1 (2017), 74–90  mathnet  crossref  crossref  mathscinet  isi  elib
    7. I. D. Shkredov, “Some remarks on sets with small quotient set”, Sb. Math., 208:12 (2017), 1854–1868  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. E. A. Yazici, B. Murphy, M. Rudnev, I. Shkredov, “Growth estimates in positive characteristic via collisions”, Int. Math. Res. Notices, 2017, no. 23, 7148–7189  crossref  mathscinet  isi
    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. G. Amirkhanyan, A. Bush, E. Croot, Ch. Pryby, “Sets of rich lines in general position”, J. Lond. Math. Soc.-Second Ser., 96:1 (2017), 67–85  crossref  mathscinet  zmath  isi  scopus
    11. A. Balog, O. Roche-Newton, D. Zhelezov, “Expanders with superquadratic growth”, Electron. J. Comb., 24:3 (2017), P3.14  mathscinet  isi
    12. A. Balog, T. D. Wooley, “A low-energy decomposition theorem”, Q. J. Math., 68:1, SI (2017), 207–226  crossref  mathscinet  zmath  isi  scopus
    13. T. Pham, M. Tait, C. Timmons, L. A. Vinh, “A Szemeredi–Trotter type theorem, sum-product estimates in finite quasifields, and related results”, J. Comb. Theory Ser. A, 147 (2017), 55–74  crossref  mathscinet  zmath  isi  scopus
    14. M. Rudnev, “On the number of incidences between points and planes in three dimensions”, Combinatorica, 38:1 (2018), 219–254  crossref  mathscinet  zmath  isi  scopus
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    16. J. Solymosi, C. Wong, “An application of kissing number in sum-product estimates”, Acta Math. Hung., 155:1 (2018), 47–60  crossref  mathscinet  isi  scopus
    17. N. Hegyvari, F. Hennecart, “Expansion for cubes in the Heisenberg group”, Forum Math., 30:1 (2018), 227–236  crossref  mathscinet  zmath  isi  scopus
    18. A. Bush, E. Croot, “Few products, many $h$-fold sums”, Int. J. Number Theory, 14:8 (2018), 2107–2128  crossref  mathscinet  zmath  isi  scopus
    19. I. D. Shkredov, “An application of the sum-product phenomenon to sets avoiding several linear equations”, Sb. Math., 209:4 (2018), 580–603  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. A. Iosevich, O. Roche-Newton, M. Rudnev, “On discrete values of bilinear forms”, Sb. Math., 209:10 (2018), 1482–1497  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. E. A. Yazici, “Sum-product type estimates for subsets of finite valuation rings”, Acta Arith., 185:1 (2018), 9–18  crossref  mathscinet  zmath  isi  scopus
    22. I. D. Shkredov, “On asymptotic formulae in some sum-product questions”, Trans. Moscow Math. Soc., 2018, 231–281  mathnet  crossref  mathscinet  elib
    23. Yu. N. Shteinikov, “On the size of the quotient of two subsets of positive integers”, Proc. Steklov Inst. Math., 303 (2018), 259–267  mathnet  crossref  crossref  mathscinet  isi  elib
    24. B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, K. S. Ryutin, I. D. Shkredov, “Sergei Vladimirovich Konyagin turns 60”, Proc. Steklov Inst. Math., 303 (2018), 1–9  mathnet  crossref  crossref  mathscinet  isi  elib
    25. C. Pohoata, “On cartesian products which determine few distinct distances”, Electron. J. Comb., 26:1 (2019), P1.7  zmath  isi
    26. O. Roche-Newton, I. Z. Ruzsa, Ch.-Y. Shen, I. D. Shkredov, “On the size of the set aa plus a”, J. Lond. Math. Soc.-Second Ser., 99:2 (2019), 477–494  crossref  zmath  isi  scopus
    27. A. Basit, B. Lund, “An improved sum-product bound for quaternions”, SIAM Discret. Math., 33:2 (2019), 1044–1060  crossref  isi
    28. B. Hanson, O. Roche-Newton, D. Zhelezov, “On iterated product sets with shifts”, Mathematika, 65:4 (2019), 831–850  crossref  isi
    29. B. Murphy, M. Rudnev, I. D. Shkredov, Yu. N. Shteinikov, “On the few products, many sums problem”, J. Theor. Nr. Bordx., 31:3 (2019), 573–602  mathscinet  isi
    30. G. Shakan, “On higher energy decompositions and the sum-product phenomenon”, Math. Proc. Camb. Philos. Soc., 167:3 (2019), 599–617  crossref  mathscinet  isi
    31. B. Hanson, O. Roche-Newton, D. Zhelezov, “On iterated product sets with shifts, II”, Algebr. Number Theory, 14:8 (2020), 2239–2260  crossref  mathscinet  isi
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