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Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

Part of: Multiplicative number theory Exponential sums and character sums Diophantine equations

Published online by Cambridge University Press:  23 February 2016

Adam J. Harper
Adam J. Harper*
Affiliation:
Jesus College, Cambridge CB5 8BL, UK email A.J.Harper@dpmms.cam.ac.uk
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Abstract

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We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that 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 $\log ^{C}x\leqslant y\leqslant x$. 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 $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

Keywords

exponential sumscircle methodrestriction theorysmooth numbersxyz conjectureRoth’s theorem

MSC classification

Primary: 11L07: Estimates on exponential sums
Secondary: 11N25: Distribution of integers with specified multiplicative constraints 11D04: Linear equations
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1121 - 1158
Copyright
© The Author 2016 

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