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Multiplicative congruences with variables from short intervals

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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.

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Author information

Authors and Affiliations

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

    Jean Bourgain

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México Morelia, 50587, Michoacán, México

    Moubariz Z. Garaev

  3. Steklovmathematical Institute, Moscow, 119991, Russia

    Sergei V. Konyagin

  4. School of Mathematics and Statistics, The University of New Southwales, Sydney, NSW, 2052, 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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Corresponding author

Correspondence to Jean Bourgain.

Additional information

The research of J. B. was partially supported by National Science Foundation Grant DMS-0808042.

The research of M. Z. G. was supported by the sabbatical grant PASPA-DGAPA-UNAM.

The research of S. V. K. was supported by Russian Fund for Basic Research Grant N. 14-01-00332 and Program Supporting Leading Scientific Schools Grant Nsh-3082.2014.1.

The research of I. E. S. was supported by Australian Research Council Grants DP1092835 and DP130100237.

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Bourgain, J., Garaev, M.Z., Konyagin, S.V. et al. Multiplicative congruences with variables from short intervals. JAMA 124, 117–147 (2014). https://doi.org/10.1007/s11854-014-0029-2

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  • DOI: https://doi.org/10.1007/s11854-014-0029-2

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