The sliding-sum method for short exponential sums

2013/06/29 by É. Fouvry, Étienne Fouvry, Emmanuel Kowalski +5 · 1 citation
Computer Science · Mathematics · #11L05 #11L07 #11T23 #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11L05 #msc:11L07 #msc:11T23

paper · pdf · doi:10.48550/arxiv.1307.0135

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

openalex publication_date 2013/06/29 · arxiv created 2015/08/04 · arxiv updated 2015/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• An inverse theorem for Gowers norms of trace functions over prime fields

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• On short sums of trace functions

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