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Saving the logarithmic factor in the error term estimates of some congruence problems

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Abstract

Combining previous ideas from Garaev and the first author, we prove a general theorem to estimate the number of elements of a subset A of an abelian group \(G=\mathbb {Z}_{n_1}\times \cdots \times \mathbb {Z}_{n_k}\) lying in a k-dimensional box. In many cases, this approach allow us to improve, by a logarithm factor, the range where it is possible to obtain an asymptotic estimate for the number of solutions of a given congruence.

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Acknowledgments

This work supported by Grants MTM 2011-22851 of MICINN and ICMAT Severo Ochoa Project SEV 2011-0087. The second author is funded by Australian Research Council Grants DP140100118.

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Authors and Affiliations

  1. Departamento de Matemáticas, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Javier Cilleruelo

  2. Department of Pure Mathematics, University of New South Wales, Sydney, 2052, Australia

    Ana Zumalacárregui

Authors
  1. Javier Cilleruelo
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  2. Ana Zumalacárregui
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Correspondence to Ana Zumalacárregui.

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Deceased: Javier Cilleruelo.

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Cilleruelo, J., Zumalacárregui, A. Saving the logarithmic factor in the error term estimates of some congruence problems. Math. Z. 286, 545–558 (2017). https://doi.org/10.1007/s00209-016-1771-1

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  • DOI: https://doi.org/10.1007/s00209-016-1771-1

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