On short sums of trace functions
Number Theory
2016-06-24 v3
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.
Cite
@article{arxiv.1508.00512,
title = {On short sums of trace functions},
author = {É. Fouvry and E. Kowalski and Ph. Michel and C S. Raju and J. Rivat and K. Soundararajan},
journal= {arXiv preprint arXiv:1508.00512},
year = {2016}
}
Comments
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