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 $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| \ge N^{\alpha} $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.