Maximal estimates for Weyl sums on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">T</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math>

In this paper, we obtain the maximal estimate for Weyl sums on torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to endpoint. We also consider two variants of problem include along rational lines and generic torus. Applications, some new upper bound Hausdorff dimension sets associated large value sums, reflect compound phenomenon between square root cancellation constructive interference. Appendix, an alternate proof Theorem 1.1 inspired by Baker's argument in [1] given Barron, improves $N^{\epsilon}$ loss 1.1, Strichartz-type estimates logarithmic losses are obtained same argument.