Weyl和的渐近性质及其相关例外集的Hausdorff维数

Weyl和是一类三角级数和，是由H.Weyl在研究序列的模一分布时引入。Weyl和在许多基本数论问题中起到重要作用，包括序列模一分布，Waring问题，Riemann zeta函数的非零区域估计等。近年来，在测度论框架下Weyl和渐近估计的研究受到了广泛关注。Weyl和渐近性质的Lebesgue测度理论与Hausdorff维数理论是当前Weyl和研究的热点问题。在本项目中我们拟研究以下三方面内容：1.估计Weyl和在Lebesgue测度意义下几乎处处的界；2.估计Weyl和例外集的Hausdorff维数；3.受Vinogradov关于素变数指数和研究的启发，我们拟引入序列上Weyl和的概念并探究其几乎处处的渐近估计。本项目的开展有助于我们建立Weyl和与Hausdorff维数的联系，加深对Weyl和及其例外集结构的理解。

Weyl sums are a class of trigonometric sums, which was introduced by H. Weyl when he investigated distribution modulo one of sequences. Weyl sums play a crucial role in many fundamental number theoretic problems, including distribution modulo one of sequences, Waring problems, and estimating the zero-free region of the Riemann zeta function etc. Recently, under the framework of the measure theoretic point of view the asymptotic properties of Weyl sums have attracted strong attention. The Lebesgue measure theory and the Hausdorff dimension theory of the asymptotic properties for Weyl sums are core issues. In this project we will study the following three parts: 1. We study the bounds of Weyl sums for almost all parameters in the sense of Lebesgue measure; 2. We estimate the Hausdorff dimension of the related exceptional sets; 3. Inspired by the work of Vinogradov on the exponential sums with prime parameters, we will introduce Weyl sums on the sequence and investigate its almost all asymptotic properties in the sense of Lebesgue measure. This project will help us to establish the connections between Weyl sums and the Hausdorff dimension, and will help us to understand the structure of Weyl sums and its related exceptional sets.