RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
Journal history

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Tr. Mosk. Mat. Obs.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 1 scientific paper (total in 1 paper)

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)
References: PDF file   HTML file

English version:
Transactions of the Moscow Mathematical Society, 2018, 231–281

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}
\elib{http://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{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060997066}


Linking options:
  • http://mi.mathnet.ru/eng/mmo616
  • http://mi.mathnet.ru/eng/mmo/v79/i2/p271

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


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

    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  isi  elib
  • Trudy Moskovskogo Matematicheskogo Obshchestva
    Number of views:
    This page:61
    Full text:8
    References:13
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019