Estimates for smooth Weyl sums on minor arcs

Abstract

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $\alpha n^k$. In particular, when $k\ge 6$ and $\rho(k)$ is defined via the relation $\rho(k)^{-1}=k(\log k+8.02113)$, then for all large numbers $N$ there is an integer $n$ with $1\le n\le N$ for which $\| \alpha n^k\|\le N^{-\rho(k)}$.