论文标题
Weyl和下限的度量理论
Metric theory of lower bounds on Weyl sums
论文作者
论文摘要
我们证明了集合$ \ mathbf {x} \ in [0,1）^d $的hausdorff尺寸，因此$ \ left | weft | \ sum_ {n = 1}^n \ exp \ left（2πi\ left（2πi\ left（2πi\ left）持有无限的许多自然数$ n $，至少为$ d-1/2d $，对于$ d \ ge 3 $，至少为$ d = 2 $的$ 3/2 $，其中$ c $仅取决于$ d $。这改善了$ d \ ge 3 $的第一和第三作者的先前下限。我们还获得了具有单一元素$ xn^d $的大笔的豪斯多夫尺寸的类似界限。
We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$, such that $$ \left|\sum_{n=1}^N \exp\left(2 πi\left(x_1n+\ldots+x_d n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d \ge 3$ and at least $3/2$ for $d=2$, where $c$ is a constant depending only on $d$. This improves the previous lower bound of the first and third authors for $d\ge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $xn^d$.
