Posted on Categories:丢番图逼近, 数学代写

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

数学代写|丢番图逼近代写DIOPHANTINE APPROXIMATION代考|Probabilistic and dimension theoretic results

After the results of the previous sections, a next natural direction is to investigate how well ‘typical’ numbers can be approximated by rationals. There are various ways to make this precise. For example we might decide to look for results which hold Lebesgue almost everywhere, or we might only require them to hold on a set of large Hausdorff dimension. In this section we will look at results of both of these types, in order to gain a more complete picture of this subject. First we have the following theorem due to Borel (1909) and Bernstein (1912).
Theorem 1.3.1. For Lebesgue almost every $\alpha \in \mathbb{R}$ we have that
$$\inf _{n \in \mathbb{N}} n|n \alpha|=0 .$$
It follows immediately from this theorem that $|\mathcal{B}|=0$ (recall that we use the notation $|A|$ to denote the Lebesgue measure of a measurable set). Equivalently, almost every $\alpha$ has unbounded partial quotients in its continued fraction expansion. Therefore, Borel and Bernstein’s theorem tells us that badly approximable numbers are not typical, in the sense of Lebesgue measure. However, it turns out that they are typical in the sense of Hausdorff dimension, as demonstrated by the following result of Jarnik (1929).
Theorem 1.3.2. The set $\mathcal{B}$ has Hausdorff dimension one.
Next, we might ask whether a result stronger than Theorem $1.3 .1$ holds, for Lebesgue almost every real number. In order to present things in a larger framework, we first make a few definitions.

数学代写|丢番图逼近代写DIOPHANTINE APPROXIMATION代考|Extensions to higher dimensions and transference principles

Here we turn to the problem of obtaining higher dimensional generalizations of our above results. Some of the arguments used in one-dimensional approximation can be adapted directly to higher dimensions. However, one of the difficulties is that there is no single expansion or multi-dimensional algorithm which does all of the things that the continued fraction expansion does in one dimension. Fortunately, for our applications in later chapters there are still tools which can be used to get around this difficulty.

Let $L: \mathbb{R}^d \rightarrow \mathbb{R}^{k-d}$ be a linear map, which is defined by a matrix with entries $\left{\alpha_{i j}\right} \in \mathbb{R}^{d(k-d)}$. For any $N \in \mathbb{N}$, there exists an $n \in \mathbb{Z}^d$ with $|n| \leq N$ and
$$|L(n)| \leq \frac{1}{N^{d /(k-d)}} .$$
This is a multidimensional analogue of Dirichlet’s Theorem, which follows from a straightforward application of the pigeonhole principle. We are interested in having an inhomogeneous version of this result, requiring the values taken by $|L(n)-\gamma|$ to be small, for all choices of $\gamma \in \mathbb{R}^{k-d}$. For this purpose we will use the following ‘transference theorem,’ a proof of which can be found in [9, Chapter V, Section 4].

数学代写|丟番图逼近代写DIOPHANTINE APPROXIMATION代考|Probabilistic and dimension theoretic results

$$\inf {n \in \mathbb{N}} n|n \alpha|=0$$ 从这个定理可以直接得出 $|\mathcal{B}|=0$ (回想一下，我们使用符号 $|A|$ 表示可测集的勒贝格测度)。等价地，几乎每一个 $\alpha$ 在其连分数 展开中有无界的偏商。因此，Borel 和 Bernstein 的定理告诉我们，在勒贝格测度的意义上，难以逼近的数字是不典型的。然而， 事实证明它们在豪斯多夫维数的意义上是典型的，正如 Jarnik (1929) 的以下结果所证明的那样。 定理 1.3.2。套装 $\mathcal{B}$ 具有豪斯多夫一维。 接下来，我们可能会问一个结果是否比定理强 $1.3 .1$ 对于勒贝格几夹所有实数都成立。为了在更大的框架下呈现事物，我们先做几 个定义。

数学代写|丟番图逼近代写DIOPHANTINE APPROXIMATION代考|Extensions to higher dimensions and transference principles

$$|L(n)| \leq \frac{1}{N^{d /(k-d)}} .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。