Jean Bourgain, Moubariz Z. Garaev, Sergei V. Konyagin, Igor E. Shparlinski, “On congruences with products of variables from short intervals and applications”, Orthogonal series, approximation theory, and related problems, Collected papers. Dedicated to Academician Boris Sergeevich Kashin on the occasion of his 60th birthday, Trudy Mat. Inst. Steklova, 280, MAIK Nauka/Interperiodica, Moscow, 2013, 67–96; Proc. Steklov Inst. Math., 280 (2013), 61

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Volume 280, Pages 67–96
DOI: https://doi.org/10.1134/S0371968513010056 (Mi tm3445)

This article is cited in 33 scientific papers (total in 33 papers)

On congruences with products of variables from short intervals and applications

Jean Bourgain^a, Moubariz Z. Garaev^b, Sergei V. Konyagin^c, Igor E. Shparlinski^d

^a Institute for Advanced Study, Princeton, NJ, USA
^b Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México
^c Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
^d Department of Computing, Macquarie University, Sydney, NSW, Australia

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DOI: https://doi.org/10.1134/S0371968513010056

Abstract: We obtain upper bounds on the number of solutions to congruences of the type

$(x_1+s)\dots(x_\nu+s)\equiv(y_1+s)\dots(y_\nu +s)\not\equiv0\pmod p$ modulo a prime

$p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.

Funding agency	Grant number
National Science Foundation	DMS-0808042
Russian Foundation for Basic Research	11-01-00329
Ministry of Education and Science of the Russian Federation	NSh-6003.2012.1
Australian Research Council	DP1092835
The research was partially supported by National Science Foundation grant DMS-0808042 (J.B.), by the Russian Foundation for Basic Research (project no. 11-01-00329, S.V.K.), by a grant of the President of the Russian Federation (project no. NSh-6003.2012.1, S.V.K.), and by Australian Research Council grant DP1092835 (I.E.S.).

Received in January 2012

English version:
Proceedings of the Steklov Institute of Mathematics, 2013, Volume 280, Pages 61–90
DOI: https://doi.org/10.1134/S0081543813010057

Bibliographic databases:

Document Type: Article

UDC: 511.3+511.524

Language: English

Citation: Jean Bourgain, Moubariz Z. Garaev, Sergei V. Konyagin, Igor E. Shparlinski, “On congruences with products of variables from short intervals and applications”, Orthogonal series, approximation theory, and related problems, Collected papers. Dedicated to Academician Boris Sergeevich Kashin on the occasion of his 60th birthday, Trudy Mat. Inst. Steklova, 280, MAIK Nauka/Interperiodica, Moscow, 2013, 67–96; Proc. Steklov Inst. Math., 280 (2013), 61–90

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Linking options:

https://www.mathnet.ru/eng/tm3445

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https://www.mathnet.ru/eng/tm/v280/p67

This publication is cited in the following 33 articles:

Christian Bagshaw, Bryce Kerr, “Lattices in function fields and applications”, Mathematika, 71:2 (2025)
Bryce Kerr, Ilya D. Shkredov, Igor E. Shparlinski, Alexandru Zaharescu, “ENERGY BOUNDS FOR MODULAR ROOTS AND THEIR APPLICATIONS”, J. Inst. Math. Jussieu, 2024, 1
Moubariz Z. Garaev, Zeev Rudnick, Igor E. Shparlinski, “On a family of sparse exponential sums”, Mathematische Nachrichten, 2024
Kerr B., Mohammadi A., “Points on Polynomial Curves in Small Boxes Modulo An Integer”, J. Number Theory, 223 (2021), 64–78
Dunn A., Kerr B., Shparlinski I.E., Zaharescu A., “Bilinear Forms in Weyl Sums For Modular Square Roots and Applications”, Adv. Math., 375 (2020), 107369
Diaz C.A., Garaev M.Z., Hernandez J., “Product of Subsets of Small Intervals and Points on Exponential Curves Modulo a Prime”, Acta Arith., 193:3 (2020), 309–319
Shparlinski I.E., “On Sums of Kloosterman and Gauss Sums”, Trans. Am. Math. Soc., 371:12 (2019), 8679–8697
Petridis G., Shparlinski I.E., “Bounds of Trilinear and Quadrilinear Exponential Sums”, J. Anal. Math., 138:2 (2019), 613–641
S. Macourt, “Visible points on exponential curves”, Bull. Aust. Math. Soc., 97:3 (2018), 353–359
M. Z. Garaev, “On congruences involving products of variables from short intervals”, Q. J. Math., 69:3 (2018), 769–778
M. Z. Garaev, “On distribution of elements of subgroups in arithmetic progressions modulo a prime”, Proc. Steklov Inst. Math., 303 (2018), 50–57
I. D. Shkredov, I. E. Shparlinski, “Double character sums with intervals and arbitrary sets”, Proc. Steklov Inst. Math., 303 (2018), 239–258
A. Ayyad, T. Cochrane, “The congruence $ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p$”, Proc. Amer. Math. Soc., 145:2 (2017), 467–477
M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43
I. D. Shkredov, I. E. Shparlinski, “On some multiple character sums”, Mathematika, 63:2 (2017), 553–560
I. E. Shparlinski, K. H. Yau, “Double exponential sums with exponential functions”, Int. J. Number Theory, 13:10 (2017), 2531–2543
I. E. Shparlinski, “Systems of congruences with products of variables from short intervals”, Bull. Aust. Math. Soc., 93:3 (2016), 364–371
J. Cilleruelo, M. Z. Garaev, “Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications”, Math. Proc. Camb. Philos. Soc., 160:3 (2016), 477–494
D. Zhelezov, “Improved bounds for arithmetic progressions in product sets”, Int. J. Number Theory, 11:8 (2015), 2295–2303
X. Shao, “Character sums over unions of intervals”, Forum Math., 27:5 (2015), 3017–3026

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

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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