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

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 14–11–00433

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

UDC: 511.178
MSC: 11B75
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} 

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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
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