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Tr. Mosk. Mat. Obs., 2018, Volume 79, Issue 2, Pages 271–334 (Mi mmo616)  

This article is cited in 11 scientific papers (total in 11 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

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 141100433


Full text: PDF file (586 kB)
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English version:
Transactions of the Moscow Mathematical Society, 2018, 231–281

Bibliographic databases:

UDC: 511.178
MSC: 11B75
Received: 23.01.2018
Revised: 25.07.2018

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://www.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}


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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. I. D. Shkredov, I. E. Shparlinski, “Double character sums with intervals and arbitrary sets”, Proc. Steklov Inst. Math., 303 (2018), 239–258  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. Mohammadi, “Improved bounds on Gauss sums in arbitrary finite fields”, Int. J. Number Theory, 15:10 (2019), 2027–2041  crossref  mathscinet  zmath  isi  scopus
    3. K. I. Olmezov, “Additive Properties of Slowly Increasing Convex Sets”, Math. Notes, 108:6 (2020), 827–841  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Thang Pham, Le Anh Vinh, “Distribution of distances in positive characteristic”, Pac. J. Math., 309:2 (2020), 437–451  crossref  mathscinet  zmath  isi  scopus
    5. N. G. Moshchevitin, I. D. Shkredov, “On a modular form of zaremba's conjecture”, Pac. J. Math., 309:1 (2020), 195–211  crossref  mathscinet  zmath  isi  scopus
    6. D. Di Benedetto, M. Z. Garaev, V. C. Garcia, D. Gonzalez-Sanchez, I. E. Shparlinski, C. A. Trujillo, “New estimates for exponential sums over multiplicative subgroups and intervals in prime fields”, J. Number Theory, 215 (2020), 261–274  crossref  mathscinet  zmath  isi  scopus
    7. N. Moshchevitin, B. Murphy, I. Shkredov, “Popular products and continued fractions”, Isr. J. Math., 238:2 (2020), 807–835  crossref  mathscinet  zmath  isi  scopus
    8. M. Rudnev, G. Shakan, I. D. Shkredov, “Stronger sum-product inequalities for small sets”, Proc. Amer. Math. Soc., 148:4 (2020), 1467–1479  crossref  mathscinet  zmath  isi  scopus
    9. S. Macourt, G. Petridis, I. D. Shkredov, I. E. Shparlinski, “Bounds of trilinear and trinomial exponential sums”, SIAM Discret. Math., 34:4 (2020), 2124–2136  crossref  mathscinet  zmath  isi  scopus
    10. I. D. Shkredov, “Some remarks on products of sets in the Heisenberg group and in the affine group”, Forum Math., 32:1 (2020), 189–199  crossref  mathscinet  zmath  isi  scopus
    11. I. D. Shkredov, “Nekommutativnye metody v additivnoi kombinatorike i teorii chisel”, UMN, 76:6(462) (2021), 119–180  mathnet  crossref
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