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On congruences with products of variables from short intervals and applications

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Abstract

We obtain upper bounds on the number of solutions to congruences of the type (x 1 + s)... (x ν + s) ≡ (y 1 + s)... (x ν + s) ≢ 0 (mod 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.

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Authors and Affiliations

  1. Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA

    Jean Bourgain

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México

    Moubariz Z. Garaev

  3. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Sergei V. Konyagin

  4. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Igor E. Shparlinski

Authors
  1. Jean Bourgain
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  2. Moubariz Z. Garaev
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  3. Sergei V. Konyagin
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  4. Igor E. Shparlinski
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Correspondence to Jean Bourgain.

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Bourgain, J., Garaev, M.Z., Konyagin, S.V. et al. On congruences with products of variables from short intervals and applications. Proc. Steklov Inst. Math. 280, 61–90 (2013). https://doi.org/10.1134/S0081543813010057

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  • DOI: https://doi.org/10.1134/S0081543813010057

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