Estimates for smooth Weyl sums on minor arcs

TL;DR

This paper develops new bounds for smooth Weyl sums on minor arcs, leading to improved results on the distribution of fractional parts of polynomial sequences for degrees six and higher.

Contribution

It introduces novel estimates for smooth Weyl sums on minor arcs, enhancing understanding of fractional parts distribution for high-degree polynomials.

Findings

For large N, there exists n with 1 ≤ n ≤ N such that ||α n^k|| ≤ N^{-ρ(k)}

New bounds improve previous estimates for Weyl sums on minor arcs

Results apply to polynomial degrees k ≥ 6

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 $\| \alpha n^k\|\le N^{-\rho(k)}$.