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Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers
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Harper, Adam (2016) Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers. Compositio Mathematica, 152 (6). pp. 1121-1158. doi:10.1112/S0010437X15007782 ISSN 0010-437X.
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Official URL: https://doi.org/10.1112/S0010437X15007782
Abstract
We investigate exponential sums over those numbers ≤ x all of whose prime factors are ≤ y . We prove fairly good minor arc estimates, valid whenever log 3 x ≤ y ≤ x 1 / 3 . Then we prove sharp upper bounds for the p -th moment of (possibly weighted) sums, for any real p > 2 and log C ( p ) x ≤ y ≤ x . Our proof develops an argument of Bourgain, showing this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of a + b = c in y -smooth integers less than x , whenever log C x ≤ y ≤ x . Previously this was only known assuming the Generalised Riemann Hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the y -smooth numbers, whenever log C x ≤ y ≤ x . This provides a deterministic set, of size ≈ x 1 − c , inside which Roth’s theorem holds.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Exponential sums, Number theory | ||||||||
Journal or Publication Title: | Compositio Mathematica | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0010-437X | ||||||||
Official Date: | June 2016 | ||||||||
Dates: |
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Volume: | 152 | ||||||||
Number: | 6 | ||||||||
Page Range: | pp. 1121-1158 | ||||||||
DOI: | 10.1112/S0010437X15007782 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 24 August 2018 | ||||||||
Date of first compliant Open Access: | 24 August 2018 | ||||||||
RIOXX Funder/Project Grant: |
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