# Perturbations of Weyl sums

@article{Wooley2015PerturbationsOW,
title={Perturbations of Weyl sums},
author={Trevor D. Wooley},
journal={International Mathematics Research Notices},
year={2015},
volume={2016},
pages={2632-2646}
}
• T. Wooley
• Published 1 March 2015
• Mathematics
• International Mathematics Research Notices
Write $f_k({\boldsymbol \alpha};X)=\sum_{x\le X}e(\alpha_1x+\ldots +\alpha_kx^k)$ $(k\ge 3)$. We show that there is a set ${\mathfrak B}\subseteq [0,1)^{k-2}$ of full measure with the property that whenever $(\alpha_2,\ldots ,\alpha_{k-1})\in {\mathfrak B}$ and $X$ is sufficiently large, then $$\sup_{(\alpha_1,\alpha_k)\in [0,1)^2}|f_k({\boldsymbol \alpha};X)|\le X^{1/2+4/(2k-1)}.$$ For $k\ge 5$, this improves on work of Flaminio and Forni, in which a Diophantine condition is imposed on… Expand
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• 2014
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