论文标题
在$ \ mathbb {t}^{d} $上对Weyl和的最大估计（用Alex Barron的附录）
Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)
论文作者
论文摘要
在本文中，我们获得了带有$ d \ geq 2 $的torus $ \ mathbb {t}^d $的Weyl总和的最大估计，这是尖锐的端点。我们还考虑了此问题的两个变体，其中包括沿理性线和通用圆环的最大估计值。应用程序包括一些与较大值相关的集合的Hausdorff尺寸上的新上限，反映了平方根取消和建设性干扰之间的复合现象。在附录中，Barron给出了受Baker论点启发的定理1.1的替代证明，Barron也提高了定理1.1中的$ N^ε$损失，并且通过同一论点获得了对数损失的Weyl type估计。
In this paper, we obtain the maximal estimate for the Weyl sums on the torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also consider two variants of this problem which include the maximal estimate along the rational lines and on the generic torus. Applications, which include some new upper bound on the Hausdorff dimension of the sets associated to the large value of the Weyl sums, reflect the compound phenomenon between the square root cancellation and the constructive interference. In the Appendix, an alternate proof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron, which also improves the $N^ε$ loss in Theorem 1.1, and the Strichartz-type estimates for the Weyl sums with logarithmic losses are obtained by the same argument.
