环面 $\mathbb{T}^{d}$ 上 Weyl 和的极大估计（附 Alex Barron 撰写的附录）
Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)
AI总结：
本文研究 $d\geq 2$ 维环面上 Weyl 和的极大估计，给出除端点外尖锐的结果，并推广到有理直线与一般环面，同时用于 Hausdorff 维数上界及带对数损失的 Strichartz 型估计。
摘要：
本文在 $d\geq 2$ 的环面 $\mathbb{T}^d$ 上获得了 Weyl 和的极大估计，该估计除端点外是尖锐的。我们还考虑了这一问题的两个变体，包括沿有理直线的极大估计以及在一般环面上的极大估计。应用包括对与 Weyl 和大值相关集合的 Hausdorff 维数给出若干新的上界，这些应用反映了平方根相消与相长干涉之间的复合现象。在附录中，Barron 受文献 [1] 中 Baker 论证的启发，给出了定理 1.1 的另一种证明；该证明还改进了定理 1.1 中的 $N^\varepsilon$ 损失，并利用同一论证得到了具有对数损失的 Weyl 和 Strichartz 型估计。
英文摘要：
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.
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