On short sums of trace functions

2015/08/03 by É. Fouvry, Fouvry, É., E. Kowalski +9 · 1 citation
Mathematics · #11L05 #11L07 #11T23 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11L05 #msc:11L07 #msc:11T23

paper · pdf · doi:10.48550/arxiv.1508.00512

v3; 20 pages; minor corrections; this paper strenghtens and extends the previous work "The sliding sum method for short exponential sums", arXiv:1307.0135, by Fouvry-Kowalski-Michel

arxiv created 2016/06/23 · arxiv updated 2016/06/24

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 "Polyá-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.

Citations

• The sliding-sum method for short exponential sums

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