|
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.
 Full text:
PDF file (586 kB)
References:
PDF file
HTML file
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}
Linking options:
http://mi.mathnet.ru/eng/mmo616 http://mi.mathnet.ru/eng/mmo/v79/i2/p271
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:
-
I. D. Shkredov, I. E. Shparlinski, “Double character sums with intervals and arbitrary sets”, Proc. Steklov Inst. Math., 303 (2018), 239–258
  -
A. Mohammadi, “Improved bounds on Gauss sums in arbitrary finite fields”, Int. J. Number Theory, 15:10 (2019), 2027–2041
 -
K. I. Olmezov, “Additive Properties of Slowly Increasing Convex Sets”, Math. Notes, 108:6 (2020), 827–841
-
Thang Pham, Le Anh Vinh, “Distribution of distances in positive characteristic”, Pac. J. Math., 309:2 (2020), 437–451
-
N. G. Moshchevitin, I. D. Shkredov, “On a modular form of zaremba's conjecture”, Pac. J. Math., 309:1 (2020), 195–211
-
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
-
N. Moshchevitin, B. Murphy, I. Shkredov, “Popular products and continued fractions”, Isr. J. Math., 238:2 (2020), 807–835
-
M. Rudnev, G. Shakan, I. D. Shkredov, “Stronger sum-product inequalities for small sets”, Proc. Amer. Math. Soc., 148:4 (2020), 1467–1479
-
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
-
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
-
I. D. Shkredov, “Nekommutativnye metody v additivnoi kombinatorike i teorii chisel”, UMN, 76:6(462) (2021), 119–180
|
| Number of views: |
| This page: | 151 | | Full text: | 28 | | References: | 19 | | First page: | 1 |
|
Terms of Use