Estimates for smooth Weyl sums on minor arcs

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-12-19 DOI:10.1112/blms.13219

Jörg Brüdern, Trevor D. Wooley

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引用次数: 0

Abstract

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $α n^{k}$ . In particular, when $k ⩾ 6$ 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 $1 ⩽ n ⩽ N$ for which $∥ α n^{k} ∥ ⩽ N^{- ρ (k)}$ .