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Weyl sums, mean value estimates, and Waring's problem with friable numbers

Volume 176 / 2016

Sary Drappeau, Xuancheng Shao Acta Arithmetica 176 (2016), 249-299 MSC: Primary 11L07; Secondary 11P05, 11N25. DOI: 10.4064/aa8448-7-2016 Published online: 17 October 2016

Abstract

In this paper we study Weyl sums over friable integers (more precisely, $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs, non-trivial upper bounds for them in minor arcs, and moreover a mean value estimate for friable Weyl sums with exponent essentially the same as in the classical case. As an application, we study Waring’s problem with friable numbers, with the number of summands essentially the same as in the classical case.

Authors

  • Sary DrappeauAix-Marseille Université, CNRS
    Centrale Marseille
    I2M UMR 7373
    13453 Marseille Cedex, France
    e-mail
  • Xuancheng ShaoMathematical Institute
    Radcliffe Observatory Quarter
    Woodstock Road
    Oxford OX2 6GG, United Kingdom
    e-mail

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