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.

中文翻译：

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Weyl 和的大值的 Hausdorff 维数

作者最近获得了具有大 Weyl 和的向量集合 $(x_1, \ldots, x_d)\in [0,1)^d$ 的 Hausdorff 维的下界，即 $$ \left | \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| \ge N^{\alpha} $$ 用于无穷多个整数 $N \ge 1$。在这里，我们获得了这些异常集的 Hausdorff 维数的上限。