Perturbations of Weyl sums

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 ${\mathfrak B}\subseteq [0,1)^{k-2}$offullmeasurewiththepropertythatwhenever$(\alpha_2,\ldots ,\alpha_{k-1})\in {\mathfrak B}$and$X$issufficientlylarge,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 ${\alpha }_k$, and the exponent of $X$ is $1-2\mathrm{/}(3k(k-1))$.