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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 alsoconsider two variants of this problem which include the maximal estimate alongthe rational lines and on the generic torus. Applications, which include somenew upper bound on the Hausdorff dimension of the sets associated to the largevalue of the Weyl sums, reflect the compound phenomenon between the square rootcancellation and the constructive interference. In the Appendix, an alternateproof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron,which also improves the $N^{\epsilon}$ loss in Theorem 1.1, and theStrichartz-type estimates for the Weyl sums with logarithmic losses areobtained by the same argument.

Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)