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

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Volume 294, Pages 87–98
DOI: https://doi.org/10.1134/S0371968516030055 (Mi tm3733)

This article is cited in 35 scientific papers (total in 36 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

Full-text PDF (214 kB) Citations (36)

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

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.

Received: March 24, 2016

English version:
Proceedings of the Steklov Institute of Mathematics, 2016, Volume 294, Pages 78–88
DOI: https://doi.org/10.1134/S0081543816060055

Bibliographic databases:

Document Type: Article

UDC: 511.178

Language: Russian

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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Linking options:

https://www.mathnet.ru/eng/tm3733

https://doi.org/10.1134/S0371968516030055

https://www.mathnet.ru/eng/tm/v294/p87

This publication is cited in the following 36 articles:

Ginny Ray Clevenger, Haley Havard, Patch Heard, Andrew Lott, Alex Rice, Brittany Wilson, “The sum-product problem for small sets”, Involve, 18:1 (2025), 165
Sayan Goswami, “Product of difference sets of the set of primes”, Proc. Amer. Math. Soc., 2023
Yu. N. Shteinikov, “Ob ekstremalnom mnozhestve chastnykh tselykh chisel”, Chebyshevskii sb., 24:4 (2023), 354–360
MISHA RUDNEV, SOPHIE STEVENS, “An update on the sum-product problem”, Math. Proc. Camb. Phil. Soc., 173:2 (2022), 411
Misha Rudnev, Ilya D. Shkredov, “On the growth rate in SL2(Fp) ${\rm SL_2}(\mathbb {F}_p)$ , the affine group and sum‐product type implications”, Mathematika, 68:3 (2022), 738
B. Xue, “Asymmetric estimates and the sum-product problems”, Acta Arith., 198:3 (2021), 289–311
Yu. N. Shteinikov, “O razmere mnozhestva proizvedeniya mnozhestv ratsionalnykh chisel”, Chebyshevskii sb., 21:1 (2020), 357–363
B. Hanson, O. Roche-Newton, D. Zhelezov, “On iterated product sets with shifts, II”, Algebr. Number Theory, 14:8 (2020), 2239–2260
I. D. Shkredov, “Any small multiplicative subgroup is not a sumset”, Finite Fields their Appl., 63 (2020), UNSP 101645
M. Rudnev, I. D. Shkredov, S. Stevens, “On the energy variant of the sum-product conjecture”, Rev. Mat. Iberoam., 36:1 (2020), 207–232
C. Pohoata, “On Cartesian products which determine few distinct distances”, Electron. J. Comb., 26:1 (2019), P1.7, 7 pp.
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
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
A. Basit, B. Lund, “An improved sum-product bound for quaternions”, SIAM Discret. Math., 33:2 (2019), 1044–1060
S. Macourt, “Decomposition of subsets of finite fields”, Funct. Approx. Comment. Math., 61:2 (2019), 243–255
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
G. Shakan, “On higher energy decompositions and the sum-product phenomenon”, Math. Proc. Camb. Philos. Soc., 167:3 (2019), 599–617
B. Liu, “Group actions, the mattila integral and applications”, Proc. Amer. Math. Soc., 147:6 (2019), 2503–2516
Ilya D. Shkredov, “Some remarks on the asymmetric sum–product phenomenon”, Moscow J. Combin. Number Theory, 8:1 (2019), 15–41
I. D. Shkredov, “Differences of subgroups in subgroups”, Int. J. Number Theory, 14:4 (2018), 1111–1134

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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