A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients $(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of $[0,1]^d$. We also use a different technique to give similar results for sums with just one monomial $xn^d$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.

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关于 Weyl 和的大值

Menshov 的一个特例——Rademacher 定理意味着对于几乎所有的多项式 $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ 的次数 $d$ 外尔和满足上限$$ \左| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty。$$ 在这里，我们研究了具有无穷多个 $N$ 的大外尔和值的特殊系数集 $(x_1, \ldots, x_d)$，并表明就贝尔类别和豪斯多夫维数而言，它们非常庞大，特别是在 $[0,1]^d$ 内的任何固定立方体中的正 Hausdorff 维数。我们还使用一种不同的技术来为只有一个单项式 $xn^d$ 的总和提供类似的结果。我们应用这些结果来表明分布不良的模一多项式的集合也相当庞大。