On short sums of trace functions

Abstract : We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Poly\'a-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give applications related to trace functions over finite fields.
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https://hal.archives-ouvertes.fr/hal-01260203
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Submitted on : Thursday, January 21, 2016 - 4:32:08 PM
Last modification on : Tuesday, March 5, 2019 - 9:30:11 AM

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  • HAL Id : hal-01260203, version 1
  • ARXIV : 1508.00512

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Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, Chandra Sekhar Raju, Joël Rivat, et al.. On short sums of trace functions. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (1), pp.423-449. ⟨http://aif.cedram.org/aif-bin/fitem?id=AIF_2017__67_1_423_0⟩. ⟨hal-01260203⟩

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