Perturbations of Weyl sums

Abstract

Write $f_k(\mathbf{\alpha };X)={\sum }_{x\le X}e({\alpha }_{1}x+\dots +{\alpha }_k{x}^k)$ $(k\ge 3)$. We show that there is a set $\mathfrak{B}\subseteq [0,1{)}^{k-2}$ of full measure with the property that whenever $({\alpha }_2,\dots ,{\alpha }_{k-1})\in \mathfrak{B}$ and $X$ is sufficiently large, then $$\underset{({\alpha }_1,{\alpha }_k)\in [0,1{)}^2}{\sup }|f_k(\mathbf{\alpha };X)|\le X^{1/2+4/(2k-1)}.$$ 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 $1-2/(3k(k-1))$.

11 pages

References in corpus (2)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• On effective equidistribution for higher step nilflows

Cited by in corpus (8)

• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• Hybrid bounds on two-parametric family Weyl sums along smooth curves
• An $L^4$ maximal estimate for quadratic Weyl sums
• Restricted mean value theorems and metric theory of restricted Weyl sums
• Effective equidistribution for generalized higher step nilflows
• On a Hybrid Version of the Vinogradov Mean Value Theorem
• Hausdorff dimension of the large values of Weyl sums
• Metric theory of lower bounds on Weyl sums