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Tr. Mat. Inst. Steklova, 2013, Volume 280, Pages 67–96 (Mi tm3445)  

This article is cited in 24 scientific papers (total in 24 papers)

On congruences with products of variables from short intervals and applications

Jean Bourgaina, Moubariz Z. Garaevb, Sergei V. Konyaginc, Igor E. Shparlinskid

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

Abstract: We obtain upper bounds on the number of solutions to congruences of the type $(x_1+s)…(x_\nu+s)\equiv(y_1+s)…(y_\nu +s)\not\equiv0\pmod p$ 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.Ż. 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.).


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

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 280, 61–90

Bibliographic databases:

Document Type: Article
UDC: 511.3+511.524
Received in January 2012
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, Tr. Mat. Inst. Steklova, 280, MAIK Nauka/Interperiodica, Moscow, 2013, 67–96; Proc. Steklov Inst. Math., 280 (2013), 61–90

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    2. B. Kerr, “Solutions to polynomial congruences in well-shaped sets”, Bull. Aust. Math. Soc., 88:3 (2013), 435–447  crossref  mathscinet  zmath  isi  elib  scopus
    3. J. Cilleruelo, I. Shparlinski, “Concentration of points on curves in finite fields”, Monatsh. Math., 171:3-4 (2013), 315–327  crossref  mathscinet  zmath  isi  elib  scopus
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    5. J. Bourgain, M. Z. Garaev, “Sumsets of reciprocals in prime fields and multilinear Kloosterman sums”, Izv. Math., 78:4 (2014), 656–707  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    8. I. E. Shparlinski, “Multiple exponential and character sums with monomials”, Mathematika, 60:2 (2014), 363–373  crossref  mathscinet  zmath  isi  elib  scopus
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    11. D. Zhelezov, “Improved bounds for arithmetic progressions in product sets”, Int. J. Number Theory, 11:8 (2015), 2295–2303  crossref  mathscinet  zmath  isi  elib  scopus
    12. X. Shao, “Character sums over unions of intervals”, Forum Math., 27:5 (2015), 3017–3026  crossref  mathscinet  zmath  isi  elib  scopus
    13. I. E. Shparlinski, “Points on varieties over finite fields in small boxes”, Scholar - a Scientific Celebration Highlighting Open Lines of Arithmetic Research, Contemporary Mathematics, 655, eds. Cojocaru A., David C., Pappalardi F., Amer. Math. Soc., 2015, 209–233  crossref  mathscinet  zmath  isi
    14. M. Munsch, I. E. Shparlinski, “Congruences with intervals and subgroups modulo a prime”, Mich. Math. J., 64:3 (2015), 655–672  crossref  mathscinet  zmath  isi  elib
    15. I. E. Shparlinski, “Systems of congruences with products of variables from short intervals”, Bull. Aust. Math. Soc., 93:3 (2016), 364–371  crossref  mathscinet  zmath  isi  elib  scopus
    16. 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  crossref  mathscinet  zmath  isi  elib  scopus
    17. 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  crossref  mathscinet  zmath  isi  scopus
    18. M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43  mathnet  crossref  crossref  isi  elib  elib
    19. I. D. Shkredov, I. E. Shparlinski, “On some multiple character sums”, Mathematika, 63:2 (2017), 553–560  crossref  mathscinet  zmath  isi  scopus
    20. I. E. Shparlinski, K. H. Yau, “Double exponential sums with exponential functions”, Int. J. Number Theory, 13:10 (2017), 2531–2543  crossref  mathscinet  zmath  isi  scopus
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    22. M. Z. Garaev, “On congruences involving products of variables from short intervals”, Q. J. Math., 69:3 (2018), 769–778  crossref  mathscinet  zmath  isi  scopus
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