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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.
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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
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\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
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This publication is cited in the following articles:
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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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