Small values of Weyl sums

Abstract

We prove that the set of $(x_1,\dots ,x_d)\in [0,1{)}^d$, such that $${\underline{\lim }}_{N\to \infty }\left|\sum _{n=1}^N\exp (2\pi i(x_{1}n+\dots +x_d{n}^d))\right|=0,$$ contains a dense ${\mathcal{G}}_{\delta }$ set in $[0,1{)}^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$\sum _{n=1}^N\exp (2\pi ix{n}^d),x\in [0,1).$$

22 pages