论文标题
Weyl总和的度量理论
Metric theory of Weyl sums
论文作者
论文摘要
我们证明存在正常数$ c $和$ c $，因此对于任何整数$ d \ ge 2 $ （x_1n+\ ldots+x_d n^d \ right）\ right）\ right | \ le c n^{1/2} $$对于无限的许多自然数字$ n $都是完全的lebesque措施。这显着改善了先前的结果，而在Hausdorff维度方面已经测量了类似的集合。 We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d\neq 4$.最后，我们获得了一般指数多项式的大值的Hausdorff维度的下限。
We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 πi \left (x_1n+\ldots+x_d n^d\right)\right) \right|\le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full Lebesque measure. This substantially improves the previous results where similar sets have been measured in terms of the Hausdorff dimension. We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d\neq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.
