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