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

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

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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, Tr. Mat. Inst. Steklova, 280, MAIK Nauka/Interperiodica, Moscow, 2013, 67–96; Proc. Steklov Inst. Math., 280 (2013), 61–90

Citation in format AMSBIB
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• https://doi.org/10.1134/S0371968513010056
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