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Kloosterman sums in residue rings

Volume 164 / 2014

J. Bourgain, M. Z. Garaev Acta Arithmetica 164 (2014), 43-64 MSC: Primary 11L05. DOI: 10.4064/aa164-1-4


We generalize some of our previous results on Kloosterman sums [Izv. Mat., to appear] for prime moduli to general moduli. This requires establishing the corresponding additive properties of the reciprocal-set $ I^{-1}=\{x^{-1}: x\in I\}, $ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun–Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general moduli.


  • J. BourgainInstitute for Advanced Study
    Princeton, NJ 08540, U.S.A.
  • M. Z. GaraevCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia 58089, Michoacán, México

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