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Trudy Mat. Inst. Steklova, 2016, Volume 294, Pages 87–98 (Mi tm3733)  

This article is cited in 27 scientific papers (total in 28 papers)

New results on sums and products in $\mathbb R$

S. V. Konyagin, I. D. Shkredov

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We improve previous sum-product estimates in $\mathbb R$; namely, we prove the inequality $\max\{|A+A|,|AA|\}\gg|A|^{4/3+c}$, where $c$ is any number less than $5/9813$. New lower bounds for sums of sets with small product set are found. We also obtain results on the additive and multiplicative energies; in particular, we improve a result of Balog and Wooley.

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


DOI: https://doi.org/10.1134/S0371968516030055

Full text: PDF file (214 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 294, 78–88

Bibliographic databases:

ArXiv: 1602.03473
UDC: 511.178
Received: March 24, 2016

Citation: S. V. Konyagin, I. D. Shkredov, “New results on sums and products in $\mathbb R$”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Trudy Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 87–98; Proc. Steklov Inst. Math., 294 (2016), 78–88

Citation in format AMSBIB
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\pages 87--98
\publ MAIK Nauka/Interperiodica
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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. Yu. N. Shteinikov, “On the product sets of rational numbers”, Proc. Steklov Inst. Math., 296 (2017), 243–250  mathnet  crossref  crossref  mathscinet  isi  elib
    2. 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
    3. 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
    4. S. Stevens, F. de Zeeuw, “An improved point-line incidence bound over arbitrary fields”, Bull. London Math. Soc., 49:5 (2017), 842–858  crossref  mathscinet  zmath  isi  scopus
    5. 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
    6. 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
    7. I. D. Shkredov, “Differences of subgroups in subgroups”, Int. J. Number Theory, 14:4 (2018), 1111–1134  crossref  mathscinet  zmath  isi  scopus
    8. S. V. Konyagin, I. D. Shkredov, “On subgraphs of random Cayley sum graphs”, Eur. J. Comb., 70 (2018), 61–74  crossref  mathscinet  zmath  isi  scopus
    9. A. Fish, “On product of difference sets for sets of positive density”, Proc. Amer. Math. Soc., 146:8 (2018), 3449–3453  crossref  mathscinet  zmath  isi  scopus
    10. A. Bush, E. Croot, “Few products, many $h$-fold sums”, Int. J. Number Theory, 14:8 (2018), 2107–2128  crossref  mathscinet  zmath  isi  scopus
    11. 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
    12. 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
    13. E. Aksoy Yazici, “Sum-product type estimates for subsets of finite valuation rings”, Acta Arith., 185:1 (2018), 9–18  crossref  mathscinet  zmath  isi  scopus
    14. I. D. Shkredov, “On asymptotic formulae in some sum-product questions”, Trans. Moscow Math. Soc., 2018, 231–281  mathnet  crossref  mathscinet  elib
    15. 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
    16. 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
    17. C. Pohoata, “On Cartesian products which determine few distinct distances”, Electron. J. Comb., 26:1 (2019), P1.7, 7 pp.  mathscinet  zmath  isi
    18. O. Roche-Newton, I. E. Shparlinski, A. Winterhof, “Analogues of the balog-wooley decomposition for subsets of finite fields and character sums with convolutions”, Ann. Comb., 23:1 (2019), 183–205  crossref  mathscinet  isi  scopus
    19. 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
    20. A. Basit, B. Lund, “An improved sum-product bound for quaternions”, SIAM Discret. Math., 33:2 (2019), 1044–1060  crossref  isi
    21. S. Macourt, “Decomposition of subsets of finite fields”, Funct. Approx. Comment. Math., 61:2 (2019), 243–255  crossref  mathscinet  zmath  isi
    22. 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  zmath  isi
    23. G. Shakan, “On higher energy decompositions and the sum-product phenomenon”, Math. Proc. Camb. Philos. Soc., 167:3 (2019), 599–617  crossref  mathscinet  zmath  isi
    24. B. Liu, “Group actions, the mattila integral and applications”, Proc. Amer. Math. Soc., 147:6 (2019), 2503–2516  crossref  mathscinet  zmath  isi
    25. Yu. N. Shteinikov, “O razmere mnozhestva proizvedeniya mnozhestv ratsionalnykh chisel”, Chebyshevskii sb., 21:1 (2020), 357–363  mathnet  crossref
    26. B. Hanson, O. Roche-Newton, D. Zhelezov, “On iterated product sets with shifts, II”, Algebr. Number Theory, 14:8 (2020), 2239–2260  crossref  mathscinet  zmath  isi
    27. I. D. Shkredov, “Any small multiplicative subgroup is not a sumset”, Finite Fields their Appl., 63 (2020), UNSP 101645  crossref  mathscinet  isi
    28. M. Rudnev, I. D. Shkredov, S. Stevens, “On the energy variant of the sum-product conjecture”, Rev. Mat. Iberoam., 36:1 (2020), 207–232  crossref  mathscinet  zmath  isi
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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