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Trudy Moskovskogo Matematicheskogo Obshchestva, 2018, Volume 79, Issue 2, Pages 271–334 (Mi mmo616)  

This article is cited in 23 scientific papers (total in 23 papers)

On asymptotic formulae in some sum-product questions

I. D. Shkredovabc

a MIPT, Institutskii per. 9, Dolgoprudnii, Russia, 141701
b Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, Russia, 119991
c IITP RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994
References:
Abstract: In this paper we obtain a series of asymptotic formulae in the sum-product phenomena over the prime field $ \mathbb{F}_p$. In the proofs we use the usual incidence theorems in $ \mathbb{F}_p$, as well as the growth result in $ \mathrm {SL}_2 (\mathbb{F}_p)$ due to Helfgott. Here are some of our applications:
  • a new bound for the number of the solutions to the equation $ (a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $ \,a_i, a'_i\in A$, $ A$ is an arbitrary subset of $ \mathbb{F}_p$,
  • a new effective bound for multilinear exponential sums of Bourgain,
  • an asymptotic analogue of the Balog–Wooley decomposition theorem,
  • growth of $ p_1(b) + 1/(a+p_2 (b))$, where $ a,b$ runs over two subsets of $ \mathbb{F}_p$, $ p_1,p_2 \in \mathbb{F}_p [x]$ are two non-constant polynomials,
  • new bounds for some exponential sums with multiplicative and additive characters.
Key words and phrases: sum-product phenomenon, asymptotic formulae, incidence geometry, exponantial sums.
Funding agency Grant number
Russian Science Foundation 14–11–00433
Received: 23.01.2018
Revised: 25.07.2018
English version:
Transactions of the Moscow Mathematical Society, 2018, Pages 231–281
DOI: https://doi.org/10.1090/mosc/283
Bibliographic databases:
Document Type: Article
UDC: 511.178
MSC: 11B75
Language: Russian
Citation: I. D. Shkredov, “On asymptotic formulae in some sum-product questions”, Tr. Mosk. Mat. Obs., 79, no. 2, MCCME, M., 2018, 271–334; Trans. Moscow Math. Soc., 2018, 231–281
Citation in format AMSBIB
\Bibitem{Shk18}
\by I.~D.~Shkredov
\paper On asymptotic formulae in some sum-product questions
\serial Tr. Mosk. Mat. Obs.
\yr 2018
\vol 79
\issue 2
\pages 271--334
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo616}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3881467}
\elib{https://elibrary.ru/item.asp?id=37045101}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2018
\pages 231--281
\crossref{https://doi.org/10.1090/mosc/283}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060997066}
Linking options:
  • https://www.mathnet.ru/eng/mmo616
  • https://www.mathnet.ru/eng/mmo/v79/i2/p271
  • This publication is cited in the following 23 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Trudy Moskovskogo Matematicheskogo Obshchestva
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