Hausdorff dimension of the large values of Weyl sums

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1,\dots ,x_d)\in [0,1{)}^d$ with large Weyl sums, namely of vectors for which $$\mid \sum _{n=1}^N\exp (2\pi i(x_{1}n+\dots +x_d{n}^d))\mid \ge N^{\alpha }$$ for infinitely many integers $N\ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.