The sliding-sum method for short exponential sums
Abstract
We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are barely insufficient to obtain non-trivial results. In particular, we prove various estimates for exponential sums over intervals in finite fields and related sums just below the Polya-Vinogradov range, and derive applications to equidistribution problems.
Cite
@article{arxiv.1307.0135,
title = {The sliding-sum method for short exponential sums},
author = {É. Fouvry and E. Kowalski and Ph. Michel},
journal= {arXiv preprint arXiv:1307.0135},
year = {2015}
}
Comments
20 pages; the basic estimate in this paper has been significantly strengthened in a joint work with CS. Raju, J. Rivat and K. Soundararajan (arXiv:1508:00512); the "sliding sum" preprint is kept separately on arXiv since it contains some results which are not discussed in the new work