Perturbations of Weyl sums

  title={Perturbations of Weyl sums},
  author={Trevor D. Wooley},
  journal={International Mathematics Research Notices},
  • 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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