小弧上光滑Weyl和的估计

IF 0.9 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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摘要

我们提供了小弧上光滑Weyl和的新估计，并探讨了它们对α n k的分数部分分布的影响$\alpha n^k$。特别是，当k小于6 $k\geqslant 6$和ρ (k) $\rho (k)$通过关系ρ定义时(k)−1 = k （log k + 8.02113）$\rho (k)^{-1}=k(\log k+8.02113)$，那么对于所有的大数N $N$，有一个整数N $n$, 1≤N≤N $1\leqslant n\leqslant N$∥α n k∥n−ρ (k)$\Vert \alpha n^k\Vert \leqslant N^{-\rho (k)}$。

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Estimates for smooth Weyl sums on minor arcs

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 $∥ α n^{k} ∥ ⩽ N^{- ρ (k)}$ .