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In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along the tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki in [8] for the case m > 1 to the case 0 < m < 1, which is sharp up to the endpoint.

In this paper, we obtain the maximal estimate for the Weyl sums on the torus T d with d ≥ 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.

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