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Journal d'Analyse Mathématique

, Volume 124, Issue 1, pp 117–147 | Cite as

Multiplicative congruences with variables from short intervals

  • Jean Bourgain
  • Moubariz Z. Garaev
  • Sergei V. Konyagin
  • Igor E. Shparlinski
Article

Abstract

Recently, several bounds have been obtained on the number of solutions of congruences of the type
$$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$
where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.

Keywords

Nontrivial Solution Asymptotic Formula Prime Divisor Algebraic Number Multiplicative Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  • Jean Bourgain
    • 1
  • Moubariz Z. Garaev
    • 2
  • Sergei V. Konyagin
    • 3
  • Igor E. Shparlinski
    • 4
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de México MoreliaMichoacánMéxico
  3. 3.Steklovmathematical InstituteMoscowRussia
  4. 4.School of Mathematics and StatisticsThe University of New SouthwalesSydneyAustralia

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